EXISTENCE OF WEAK SOLUTIONS TO SOME ... - Semantic Scholar
SIAM J. MATH. ANAL. Vol. 34, No. 6, pp. 1279–1299
c 2003 Society for Industrial and Applied Mathematics
EXISTENCE OF WEAK SOLUTIONS TO SOME VORTEX DENSITY MODELS∗ QIANG DU† AND PING ZHANG‡ Abstract. We study the weak solutions to equations arising in the modeling of vortex motions in superfluids such as superconductors. The global existence of measure-valued solutions is established with a bounded Radon measure as initial data. Moreover, we get a local space-time Lq estimate for the continuous part of the solution, and we prove the global existence of a distributional weak solution for a particular case. We also consider a modification to the model in order to physically account for the different signs of vortices, and we present, in one space dimension, the global existence of weak solutions with the initial data in BV for the modified model. Key words. quantized vortices, vortex density, hydrodynamics, vortex sheets, weak convergence, measure-valued solutions AMS subject classifications. 35Q55, 35B, 35K PII. S0036141002408009
1. Introduction. In this paper, we study the concentration phenomenon of the approximate solution sequences to the equations ∂t ρ + div(uρ) = 0, (t, x) ∈ (0, ∞) × R2 , u = M ∇−1 ρ, (1.1) ρ|t=0 = ρ0 , with ρ0 being a bounded Radon measure and M being a constant orthogonal matrix of the form cosθ −sin θ M (θ) = . sin θ cos θ Our investigation yields the global existence of a measure-valued solution to (1.1) and the classical weak solution to (1.1) if ρ0 is a bounded positive (resp., negative) Radon measure when cos θ > 0 (resp., cos θ < 0). For the case cos θ = 1, our results here extend those available in the literature (see, for instance, [21]). In the more general case, our study is related to the mathematical study of incompressible fluids as well as the vortex state in superfluids. Indeed, when cos θ = 0, (1.1) is the classical two-dimensional incompressible Euler equations, which can be rewritten in the velocity formulation ∂t u + div(u ⊗ u) = −∇P, (t, x) ∈ (0, ∞) × R2 , divu = 0, (1.2) u|t=0 = u0 . ∗ Received by the editors May 21, 2002; accepted for publication (in revised form) November 18, 2002; published electronically May 12, 2003. This work was supported in part by the Chinese NSF, innovation grants from the Chinese Academy of Sciences, the state key basic research project G199903280, and the U.S. NSF grant DMS-0196522. This work was completed during Ping Zhang’s visit to Penn State University. http://www.siam.org/journals/sima/34-6/40800.html † Department of Mathematics, Penn State University, University Park, PA 16802, and Lab for Scientific and Engineering Computing, CAS, China ([email protected]). ‡ Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China ([email protected]).
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The initial value problem to (1.2) with u0 = (u10 , u20 ) ∈ L2loc (R2 ) and ω0 = ∂2 u10 − ∂1 u20 ∈ M(R2 ) is an outstanding open problem, known as the vortex sheets problem, in incompressible fluid mechanics (see [25]). In 1991, Delort [7] proved the global existence of weak solutions to this problem when ω0 is a bounded Radon measure without negative singular part. Later, Majda [26] obtained the same result by the vanishing viscosity limit to the two-dimensional incompressible Navier–Stokes equations. When ω0 is a general Radon measure, only a measure-valued solution seems possible for this problem; see [8, 9, 27, 28] for more details. In recent years, studies of the stability, dynamics, and interactions of the vortices in both classical fluids and superfluids have received a lot of attention. For example, in the mezoscale Ginzburg–Landau models of superconductors [14], the individual vortices are resolved and their interactions and dynamics are studied in great detail. On a macroscopic level, when the number of vortices becomes exceedingly large, it is advantageous to model the vortex state using a vortex density function [4, 16]. When cos θ = 1, (1.1) has been obtained as the hydrodynamic limit of Ginzburg–Landau vortices governing by gradient dynamics. A formal derivation was given in [16], and a rigorous justification was given in [21]. Studies in this direction also include [2, 4, 15] on model derivation, [19, 29, 30] on mathematical analysis, and [5, 12, 13, 18] on numerical simulations (see [3] for additional references). The general case of θ = 0 corresponds to a complex time relaxation in the gradient dynamics. In [21], the global existence of weak solutions to (1.1) with cos(θ) = 1 and ρ0 being a positive bounded Radon measure was established. In the case of ρ0 taking on different signs, the notion of weak solutions to (1.1) with cos(θ) = 0 requires further study, as additional difficulties do arise. A similar but somewhat modified version of (1.1) was studied in [2, 4] when the vortices are of different signs. Taking the London approximation to the induced magnetic field into account, a system of equations similar to (1.1) with cos(θ) = 1 was derived in [4]: ∂t ρ + div(u|ρ|) = 0, (t, x) ∈ (0, ∞) × R2 , u = ∇(λ2 − I)−1 ρ, (1.3) ρ|t=0 = ρ0 . Here, λ denotes the penetration depth. The density function ρ is allowed to change sign in order to represent vortices of different signs. The general case of θ = 0 can also be easily derived when the time relaxation parameter becomes complex valued [17]. In [11], such an approach was taken to account for the Hall effect. A vector-valued version of (1.3) was also available [4] to account for the three-dimensional effect. The existence and uniqueness of a viscosity solution to an equation similar to (1.3) was proved in [19] by the viscosity solution method in [6] for an R2 -valued function ρ, since a scalar stream function ψ can be found in that case such that ρ = ∇⊥ ψ. Such a technique obviously is not applicable here. To our knowledge, the general existence to (1.3) is still open except for the stationary solutions studied in [4]. To draw an analogy with (1.1), we consider a modification of the above equation: ∂t ρ + div(u|ρ|) = 0, (t, x) ∈ (0, ∞) × R2 , u = ∇−1 ρ, (1.4) ρ|t=0 = ρ0 . Notice that when ρ0 ≥ 0, (1.4) is the same as (1.1).
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This paper consists of two main parts. In the first part, we study the global existence of weak solutions to (1.1) with general Radon measure as initial data under the condition that cos θ = 0 and obtain more general results than those given in [21]. Without loss of generality, we restrict ourselves to the case cos θ > 0; the results for cos θ < 0 can be similarly obtained. In the second part of the paper, we present an existence result to (1.4) in one space dimension. To introduce our main results, let us examine the general procedure on proving the global existence of weak solutions. As in [9], the first step is to construct the approximate solution sequences. For simplicity, let us define the following cut-off function: ξ, ξ ≥ −1/, T (ξ) := (1.5) −1/, ξ ≤ −1/. We study the approximate solution sequence to (1.1) constructed by the equations ∂t ρ + u ∇ρ = −cos θT (ρ )ρ , (t, x) ∈ R+ × R2 , u = M (θ)∇−1 ρ , (1.6) ρ |t=0 = ρ0, , 1, |x| ≤ 1, where ρ0, = (ρ0 χ ) ∗ j , χ (x) = χ( x ), χ ∈ Cc∞ (R2 ), χ(x) = 0, |x| ≥ 2, and j (x)
is the standard Friedrich’s mollifier with suppj ⊂ B (0). Let S (ξ) = |ξ| ∗ j . The approximate solution sequence to (1.4) may be defined by the following equation: ∂t ρ + div(u S (ρ )) = ρ , (t, x) ∈ R+ × R2 , u = ∇−1 ρ , (1.7) ρ |t=0 = (ρ0 χ ) ∗ j . Now, a main result of this paper can be stated in the following theorem. Theorem 1.1. Let ρ0 ∈ M(R2 ) and cosθ > 0. Then there exist a subsequence of {ρ , u } constructed by (1.6) (still denoted by {ρ , u } for convenience), functions 1,q ρ ∈ Lqloc (R+ × R2 ) ∩ L∞ (R+ , L1 (R2 )) and u ∈ Lqloc (R+ , Wloc (R2 )) for any q < 2, and + + 2 a positive Radon measure µ ∈ M (R × R ) such that the following hold: 1. The following convergence properties and estimates hold: (1.8) (1.9) (1.10) (1.11)
ρ ρ weakly in Lqloc (R+ × R2 ), 1,q u u weakly in Lqloc (R+ , Wloc (R2 )), 1 ρ 1ρ ≤− 1 µ in the sense of ρ + ∞ dµ ≤ | dρ0 (x)|. 0
R2
M(R+ × R2 ),
R2
2. The following decay estimates hold: ρ(t, x) ≤
(1.12)
cos θ t
for a.e.
(t, x) ∈ R+ × R2 .
3. For all test functions ϕ ∈ Cc∞ ([0, ∞) × R2 ), there holds ∞ (1.13) (ρ∂t ϕ + ρu∇ϕ + µϕ) dx dt + ϕ(0, x)ρ0 dx = 0 0
R2
R2
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and u = M (θ)∇−1 ρ.
(1.14)
Definition 1.1. We call (ρ, u, µ) the measure-valued solution to (1.1) if (ρ, u, µ) satisfies (1.13) and (1.14). A similar definition was used by Alexandre and Villani in the study of the Boltzmann equation without Grad’s angular assumption to the cross section [1]. There, f (t, x, v) was defined as a renormalized solution to the Boltzmann equation with a defect measure µ(t, x, v) if, for all nonlinearity β ∈ C 2 (R+ , R+ ) satisfying β(0) = 0, C 0 < β (f ) ≤ 1+f , β (f ) < 0, there holds ∂t β(f ) + v · ∇β(f ) = β (f )Q(f, f ) + µ in the sense of distributions. One may check [1] for more details. When ρ0 ∈ M+ (R2 ), we have the following improvement of Theorem 1.1. Corollary 1.1. Let 0 < cos θ < 1 and ρ0 ∈ M+ (R2 ). Then µ = 0 in (1.13), and ρ ∈ L2 ((0, ∞) × R2 ). Moreover, (1.15)
ρL2 ((0,∞)×R2 ) ≤ Cρ0 (R2 ).
If 0 ≤ ρ0 ∈ Lp (R2 ) for 1 < p < ∞, we have the following better estimate for ρ: T ρp (T, x) dx+(p−1)cos θ ρp+1 (t, x) dx dt ≤ ρp0 (x) dx for a.e. T ∈ R+ . R2
(1.16)
0
R2
R2
Remark 1.1. In comparison with the measure-valued solutions to (1.2) in [8] and that of the one-dimensional two-component Vlasov–Poisson equations in [27, 28], the measure-valued solution to (1.1) here is much more explicit and closer to the distributional weak solution of (1.1). By (1.10), if ρ0 is a sign-changing Radon measure, µ may not be 0 even if {ρ } strongly converges to ρ in Lqloc (R+ × R2 ) for any q < 2. Moreover, from the corollary, we can see that if cos θ > 0 and ρ0 is a positive Radon measure, then the approximate solutions satisfy ρ (t, x) ≥ 0 for all (t, x) ∈ R+ × R2 , which in turn implies, by the definition of µ, that µ = 0. Thus, (ρ, u) is the classical distributional weak solution to (1.1). 2 Remark 1.2. If cos θ > 0 and 0 ≤ ρ0 (x) ∈ L∞ comp (R ), following exactly the same procedure as that in [21] and [33], we can prove the uniqueness of the weak solution in the above corollary. Remark 1.3. We can replace the second equation in (1.1) by u = −M (θ)∇(−λ2 + −1 1) ρ in correspondence to the equations derived in [4]. Step-by-step modifications of the proofs given here will yield similar results for such equations as those in Theorem 1.1. In one space dimension, (1.4) takes on the following form: ∂t ρ + ∂x (u|ρ|) = 0, (t, x) ∈ (0, ∞) × R, x u = −∞ ρ(t, y) dy, (1.17) ρ|t=0 = ρ0 . Then we have the following existence result to (1.17). Theorem 1.2. Let ρ0 ∈ BV (R). Then the weak limit (ρ, u) obtained by the vanishing viscosity of (1.7) satisfies (1.17) in the sense of distributions, and
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ρ ∈ C([0, T ], Lp (K)) ∩ L∞ ([0, T ], BV (R)), u ∈ L∞ ([0, T ], W 1,p (R)) for any T < ∞, 2 < p < ∞, and any compact subset K of R. Furthermore, (1.18) |ρ(t, x)| dx ≤ |ρ0 | dx, | dρ(t, x)| ≤ | dρ0 |,
R
R
R
R
where R | dρ| denotes the total variation of ρ, if supp ρ0 ⊂ Br (0), supp ρ(t, ·) ⊂ {x : |x| ≤ r + M t} with M the L∞ bound of u. Remark 1.4. (1) An explicit solution to (1.17) may be constructed: let ρ0 be defined by ρ0 (x) = 1 in (0, 1), ρ0 (x) = −1 in (1, 2), and ρ0 (x) = 0 elsewhere; then a global weak solution to (1.17) may be defined by 1 x ∈ (0, 1), 1+t , −1 x ∈ (1, 2), ρ(t, x) = 1+t , 0 otherwise. Here, x = 1 is the shock line of this solution. Naturally, this leads to a conjecture that shocks will be formed for d = 2 when ρ0 changes sign. In general we cannot prove |ρ | m = |ρ|. Nevertheless, we can prove that ∂t ρ + div(um) = 0,
u = ∇−1 ρ
holds in the sense of distributions and m = |ρ| for almost all (t, x) in a subset of R+ × R2 ; see Proposition 3.1 for more details. (2) With d = 2, to get the uniform L∞ ([0, T ], L1 (R2 )) estimate for {∂x ρ }, the solution sequence to (1.7), we need the uniform L∞ estimate for ∇u (see the proof of Lemma 3.2 for details). But with ρ being uniformly bounded, we cannot get the desired estimate for ∇ ⊗ u = ∇ ⊗ ∇−1 ρ by the singular integral operator theory [31]. However, in the case of d = 1, by the second equation of (1.17), we find ∂x u = ρ , which gives the desired estimate for ∂x u . Remark 1.5. Again, one may replace the second equation of (1.17) by u = −∂x (−λ2 ∂xx + 1)−1 ρ and, using the same type of arguments as those in Theorem 1.2, prove similar results. We now introduce some notation that will be used throughout the paper. We let Br (0) = {x : |x| ≤ r} and denote by M(Ω) the bounded Radon measure space on Ω, by M+ (Ω) the bounded positive Radon measure space on Ω, and BV (R) = {f : f ∈ L1 (R), ∂x f ∈ M(R)}. We use C(a, b, . . .) as a uniform constant which only depends on the listed variables and may change from line to line. The proofs of the above theorems are given in later sections. 2. Proof of Theorem 1.1. Now let us first prove the global existence of solutions to (1.6). For convenience, we omit the subscript in the approximate solution sequence {(ρ , u )} in the following lemma. Lemma 2.1 (solution of (1.6) with smooth data). For ρ0 ∈ Cc∞ (R2 ), (1.6) has a global strong solution (ρ, u) such that ρ ∈ L∞ ([0, T ], W 1,p (R2 )), ∇u ∈ L∞ ([0, T ], W 1,p (R2 )) for any p > 1, T < ∞, and (2.1)
ρ(t, ·)L1 ≤ ρ0 L1
and
ρ(t, x) ≤
cos θ , t
t > 0.
Proof. 1. (Blow-up principle.) Following the standard argument for a nonlinear hyperbolic equation, we can prove the local existence of solution (ρ, u) to (1.6) with
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QIANG DU AND PING ZHANG
smooth data such that ρ, ∇u ∈ L∞ ([0, T ], W 1,p (R2 )) for some positive constant T and any p < ∞. Now, let T ∗ be the lifespan of the solution (ρ, u) to (1.6). We are going to show that if T ∗ < ∞, (2.2)
lim ρ(t, ·)L∞ = ∞.
t→T ∗
In fact, for any even positive number p, it follows from (1.6) that (2.3)
∂t (∂xi ρ)p + u∇(∂xi ρ)p + p∂xi u∇ρ(∂xi ρ)p−1 = −pcos θ∂xi (T (ρ)ρ)(∂xi ρ)p−1 .
Noticing that divu = cos θρ, |∂xi (T (ρ)ρ)| ≤ 2|ρ||∂xi ρ|, integrating the above equation over R2 , and using integration by parts, we get d (2.4) |∂xi ρ|p dx ≤ ((2p + 1)cos θρL∞ + p∂xi uL∞ ) |∇ρ|p dx. dt R2 R2 Let us take χ(ξ) ∈ Cc∞ (R2 ), χ(D) the corresponding pseudodifferential operator with symbol χ(ξ); then by singular integrals theory [31, 32], we have (2.5)
χ(D)∇ ⊗ ∇−1 ρL∞ ≤ C∇ ⊗ ∇−1 ρLp ≤ CρLp .
While by Lemma B.1.C of [32], for all p > 2, we find ρW 1,p −1 (1 − χ(D))∇ ⊗ ∇ ρL∞ ≤ CρL∞ 1 + log (2.6) . ρL∞ Summing up the second equation of (1.6) and inequalities (2.5) and (2.6), we find ρW 1,p ∇uL∞ ≤ C ρL∞ 1 + log (2.7) + ρLp . ρL∞ A simple interpolation result gives us (2.8)
p−1
1
p ρLp ≤ ρL∞ ρLp 1 .
Summing up (2.4) and (2.7)–(2.8) we obtain 1 d ∇ρLp p ∇ρ(t, ·)Lp ≤ C ρL∞ 1 + log + ρL1 ∇ρpLp . (2.9) 2 dt ρL∞ Then the Gronwall inequality yields that (2.10)
∇ρ(t, ·)Lp ≤ C(T, ρL1 , ρL∞ )∇ρ0 Lp .
On the other hand, by multiplying signρ on both sides of (1.6), we find by (1.5) that (2.11)
∂t |ρ| + div(u|ρ|) = cos θ(ρ − T (ρ))|ρ| ≤ 0.
Integrating (2.11) over R2 , we get the first inequality of (2.1). Summing up (2.1) and (2.10), we complete the proof of the claim (2.2). 2. (Estimate of ρL∞ .) By (2.7) and the classical theory on ordinary differential equations, the equation dΦt (x) = u(t, Φt (x)), dt (2.12) Φt (x)|t=0 = x
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
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has a unique solution Φt (x) ∈ C([0, T ∗ ) × R2 ), and ∂x Φt (x) ∈ L∞ ([0, T ] × R2 ) for any T < T ∗ . Then, by the first equation of (1.6), we have dρ(t, Φt (x)) ≤ 0, dt which implies that ρ(t, ·) ≤ ρ0 L∞ .
(2.13)
This together with (1.5) shows that (2.14)
T (ρ)L∞ ≤ max
1 , ρ0 L∞
.
Thus by the first equation of (1.6) and by (2.12), we have d|ρ|(t, Φt (x)) 1 ∞ |ρ|(t, Φt (x)), ≤ cos θ max , ρ0 L dt which together with the Gronwall inequality yields that 1 , ρ0 L∞ t ρ0 L∞ . (2.15) ρL∞ ≤ exp cos θ max Summing up (2.2) and (2.15), we get the global existence of strong solutions to (1.6) with smooth initial data. 3. (Decay estimate.) By the first equation of (1.6) and by (2.12), we have (2.16)
dρ(t, Φt (x)) = −cos θ(T (ρ)ρ)(t, Φt (x)), dt
which implies that if ρ0 (x) ≤ 0, then ρ(t, Φt (x)) ≤ 0, and if ρ0 (x) ≥ 0, then ρ(t, Φt (x)) ≥ 0. So to prove the one-sided decay estimate (2.1), we need to consider only the points where ρ0 (x) > 0. By (1.5), T (ρ(t, Φt (x)) = ρ(t, Φt (x)). Solving (2.16), we get (2.17)
ρ(t, Φt (x)) =
cos θ cos θρ0 (x) < , 1 + tρ0 (x) t
t > 0.
Summing up the above, we get the second inequality of (2.1). This completes the proof of the lemma. Next let us get the key uniform space-time estimate for the approximate solution sequence {ρ } constructed in Lemma 2.1. Lemma 2.2 (L1+α estimate). Let ρ0, ∈ L1 (R2 ), α ∈ (0, 1), T, R > 0. Then for the solutions (ρ , u )>0 of (1.6), there holds the estimate T |ρ |1+α dx dt ≤ Cα,T,R , (2.18) 0
|x|≤R
where the constant Cα,T,R depends only on the L1 norm of ρ0, and the listed variables. Proof. 1. (Elementary estimate.) We first assume α = d2 /d1 ∈ (0, 1/2), where d1 and d2 are odd positive integers. Let χ ∈ Cc∞ (R2 ), χ ≥ 0 and χ = 1 on {x||x| ≤ R},
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QIANG DU AND PING ZHANG
ξ with supp χ ⊂ {x||x| ≤ R + 1}. Set η(ξ) = α 0 max(1, |s|)α−1 ds for ξ ∈ R1 such that η (ξ) = α max(1, |ξ|)α−1 . Multiplying (1.6) by χη (ρ ), integrating the resulting identity over [0, T ] × R2 , and performing integration by parts several times, we obtain T (2.19) cos θ χ(ρ η(ρ ) − ρ T (ρ )η (ρ )) dx dt R2
0
=
R2
χη(ρ ) dx|T0
−
0
T
R2
∇χu η(ρ ) dx ds.
First, by the second equation of (1.6), for all test functions φ(x), ψ(x) ∈ Cc∞ (R2 ), we have x−y (ψ(x) − ψ(y)) ρ (t, y) dy φ(x)ψ(x)u (t, x) = M (θ)φ(x) |x − y|2 2 R x−y φ(x)ψ(y) ρ (t, y) dy. (2.20) + M (θ) |x − y|2 R2 x−y x−y ∞ Notice that φ(x)ψ(y) |x−y| 2 = φ(x)ψ(y)ζ(|x − y|) |x−y|2 , where ζ(z) ∈ Cc (R) with ζ(z) = 1 for z ∈ suppφ + supp ψ. And trivially ζ(z) |z|z 2 ∈ Lp (R2 ) for all p < 2, by the Hausdorff–Young inequality to the second term of (2.20), we get
φψu (t, ·)Lp ≤ (sup |∇ψ|φLp + cφ,ψ )ρ (t, ·)L1
for all 1 ≤ p < 2.
In particular, by taking φ(x) = ψ(x) = 1 for |x| ≤ R + 1, we get
p1 (2.21)
|x|≤R+1
|u (t, x)|p dx
≤ CR ρ0, L1
for all 1 < p < 2.
Note that α < 12 , and thus by (2.1) and (2.21), we have
1−α
α T T 1 ∇χu η(ρ ) dx ds ≤ |u | 1−α dx |ρ | dx dt 0 R2 0 |x|≤R+1 |x|≤R+1 ≤ C1 (R, ρ0, L1 ) .
(2.22)
By (2.1) and the definition of η, we have χη(ρ )(T, x) dx 2 R ≤ χ(α + |ρ |α )(T, x) dx + α (2.23)
|ρ |≥1
2
≤ 2πα(R + 1) +
|x|≤R+1
|ρ |≤1
χ(x)|ρ | dx
α
|ρ |(T, x) dx
(2π(R + 1)2 )1−α + α2π(R + 1)2
≤ C2 (α, R, ρ0, L1 ). Finally it follows from the definition of α and η that T χ(x)(ρ η(ρ ) − ρ T (ρ )η (ρ )) dx dt 0
(2.24)
=
0
≥
0
R2 T
T
R2
R2
1|ρ |≥1 χ (1 − α)ρ1+α + α(ρ1+α − T (ρ )ρα ) + αρ dx 1|ρ |≥1 χ((1 − α)ρ1+α + αρ ) dx,
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where 1|ρ |≥1 is the characteristic function on the set {(t, x) : |ρ (t, x)| ≥ 1}. Summing up (2.19)–(2.24), we find (2.25)
T
0
χρ1+α
|ρ |≥1
1 dx dt ≤ 1−α
α
R2
χ|ρ | dx + C1 + C2
for all α = d2 /d1 ∈ (0, 1/2). 2. (Inductive step 1.) Next, we are going to show by the bootstrap method that (2.25) holds for all α ∈ (0, 1). First, let us take α = d2 /d1 ∈ (0, 5/6) with d1 , d2 positive odd integers. In particular, due to the arbitrariness of R, by interpolation, and by (2.25), we have (2.26)
T
0
|x|≤R+1
|ρ |p1 dx dt ≤ C(R, T, ρ0, )
for all p1
1, again by [20] and (2.37), we have
ρp (T, x) dx + (p − 1)cos θ 2 R ≤ lim→0 ≤ lim→0
(2.56)
R2
R2
T
0
R2
ρp+1 dx dt
ρp (T, x) dx
ρp0, (x) dx =
+ (p − 1)cos θ
R2
T
0
R2
ρp+1
dx dt
ρp0 dx.
Then, by summing (2.55) and (2.56), we complete the proof of Corollary 1.1. 3. Proof of Theorem 1.2 and the remarks. Again the first step in the proof of Theorem 1.2 is to construct the approximate solution sequence {(ρ , u )}. By Theorem A.1, we immediately have the following lemma. Lemma 3.1 (solution of (1.7) with smooth data). Let d = 1, 2, ρ0 ∈ L∞ (Rd ). Then (1.7) has a global smooth solution (ρ , u ) such that ρ , ∇u ∈ L∞ ([0, T ], H s (Rd ))∩ L2 ([0, T ], H s+1 (Rd )) for any s > d2 + 1, T < ∞, and (3.1) (3.2)
ρ (t, ·)L1 ≤ ρ0 L1 , ρ (t, ·)L∞ ≤ ρ0 L∞ + , t 2 2 2 2 ρ (t, x) dx + 2 |∇ρ | dx ds ≤ ρ0 dx + t Rd
0
Rd
Rd
Rd
1
|ρ0 | dx. 1− 1
Furthermore, if supp ρ0 ⊂ Br (0), we denote M = (ρ0 L∞ + 1) d ρ0 L1 d ; then for r > and (t, x) ∈ Ωo =: {(t, x) : |x| ≥ r + M t, t ≥ 0}, there holds (3.3)
|ρ (t, x)| ≤ ρ0 L∞ exp[−1 (r + M t − |x|) + tρL∞ ] ≡ Q (t, x).
Proof. First by the third equation of (1.7), the global existence and uniqueness of solution to (1.7) is a direct consequence of Theorem A.1. Moreover, (A.2) and (A.3) imply (3.1) and (3.2), respectively. Consequently, (A.15) implies that u L∞ ≤ M. Next, we rewrite the first equation of (1.7) as (3.4)
∂t ρ + u S (ρ )∇ρ + S (ρ )ρ = ρ .
Let us denote L = ∂t + u S (ρ )∇ + S (ρ ) − . Notice by the definition of S (ξ) that |S (ρ )| ≤ 1 and S (ρ )L∞ ≤ ρ L∞ . We find (3.5) (3.6)
LQ ≥ 0 on Ωo , Q ||x|=r+M t ≥ ρ0 L∞ ,
Q |t=0,|x|≥r ≥ 0.
Hence by the maximum principle, we have (3.7)
ρ (t, x) ≤ Q (t, x)
for all (t, x) ∈ Ωo .
Similarly, we can prove (3.8)
−ρ (t, x) ≤ Q (t, x)
for all (t, x) ∈ Ωo .
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QIANG DU AND PING ZHANG
Combining (3.7) with (3.8), we get (3.3). This completes the proof of the lemma. With this lemma, by (1.7), (3.1), (3.2), and [31], we have that ∂t u = ∂t ∇−1 ρ = −∇−1 div(u S (ρ )) + ∇ρ
(3.9)
is uniformly bounded in L∞ ([0, T ], L2 (Rd )). While again by (1.7), (3.1), and [31], we have is uniformly bounded in L∞ ([0, T ], W 1,p (Rd ))
u
(3.10)
for any 2 < p < ∞. Thus by combining (3.9), (3.10), the Lions–Aubin lemma, and the similar proof of Lemma 3 in [34], we obtain that there exists a subsequence of {u }, which we still denote by {u }, and some u ∈ L∞ ([0, T ], W 1,p (Rd )) such that u → u
(3.11)
uniformly on any compact subset of [0, T ] × Rd .
Trivially by (3.1), there exist ρ, m ∈ L∞ ([0, T ] × Rd ) such that ρ ρ weakly ∗ in L∞ ([0, T ] × Rd ), |ρ | m weakly ∗ in L∞ ([0, T ] × Rd ).
(3.12) (3.13)
While by the definition of S (ξ), for any compact subset K of [0, T ] × R2 , we have S (ρ ) − |ρ |L1 (K) → 0
(3.14)
as
→ 0.
Thus by combining the first equation of (1.7) with (3.11)–(3.14), we have that ∂t ρ + div(um) = 0
(3.15)
holds in the sense of distributions. Summing up the second equation of (1.7) and (3.11) and (3.12), we get u = ∇−1 ρ .
(3.16)
Thus, in order to prove that (ρ, u) is indeed a global weak solution to (1.4), we only need to prove that m = |ρ|. However, only in one space dimension, and ρ0 ∈ BV (R), we can prove that d(t, x) = |ρ|(t, x) for almost all (t, x) ∈ R+ × R. In order to do so, let us first present the following lemma. Lemma 3.2. Let ρ0 ∈ BV (R). Then (3.17) |∂x ρ (T, x)| dx ≤ e3T | dρ0 |,
R
R
where R | dρ0 | is the total variation of ρ0 . Proof. Let g be the solution of the adjoint equation (3.18)
∂t g + u S (ρ )∂x g − (S (ρ ) + ρ S (ρ ))g + ∂xx g = 0,
with the Cauchy data g(T, ·) = γ ∈ C0∞ ({x : |x| < R}), and γL∞ ≤ 1. Let τ = T −t, h = e−3τ g; then by (3.18), we have (3.19)
∂τ h − u S (ρ )∂x h + (S (ρ ) + ρ S (ρ ) + 3)h − ∂xx h = 0.
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1293
We assume that h reaches its minimum value at (τ0 , x0 ). Then we claim that either h(τ0 , x0 ) ≥ 0
(3.20)
or τ0 = 0.
Otherwise, if h(τ0 , x0 ) < 0 and τ0 > 0, we have by the definition of (τ0 , x0 ) that (3.21)
∂t h(τ0 , x0 ) = 0,
∂x h(τ0 , x0 ) = 0,
∂xx h(τ0 , x0 ) ≥ 0,
which implies that {∂τ h − u S (ρ )∂x h + (S (ρ ) + ρ S (ρ ) + 3)h − ∂xx h} (t0 , x0 ) < 0 as S (ρ ) ≥ 0, ρ S (ρ )+3 > 0. This contradicts (3.19), which proves the claim (3.20). Hence h(τ, x) ≥ min (0, min(h(0, x))) = −1, which implies that (3.22)
g(t, x) ≥ −e3(T −t)
for all (t, x) ∈ [0, T ] × R.
Exactly as in the proof of (3.22), we can also prove that (3.23)
g(t, x) ≤ e3(T −t)
for all (t, x) ∈ [0, T ] × R.
Combining (3.22) with (3.23), we get g(t, ·)L∞ ≤ e3(T −t) ,
(3.24)
0 ≤ t ≤ T.
With (3.24) and the similar proof to (3.3), we get (3.25)
|g(t, x)| ≤ exp[−1 (R + M (T − t) − |x|) + 4(T − t)ρ L∞ ]
for all (t, x) ∈ {(t, x) : |x| ≥ R + M (T − t), 0 ≤ t ≤ T }. On the other hand, taking ∂x to the first equation of (1.7), we have (3.26) ∂t (∂x ρ ) + ∂x (u S (ρ )∂x ρ ) + (S (ρ ) + ρ S (ρ ))∂x ρ − ∂xx ∂x ρ = 0. Combining (3.18), (3.25), and (3.26), we get ∂x ρ (T, x)γ(x) dx − ∂x ρ0, g(0, x) dx R
(3.27)
=
0
T
R
R
(∂t (∂x ρ )g + ∂x ρ ∂t g) dx dt = 0.
It follows from (3.24) that ∂x ρ (T, x)γ(x) dx ≤ e3T |∂x ρ0, | dx 2 R R R ≤ e3T (3.28) |dρ0 |, R
which implies (3.17). This completes the proof of the lemma. We now complete the proof of Theorem 1.2.
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QIANG DU AND PING ZHANG
Proof of Theorem 1.2. First by (3.17), we find that {ρ (t, x)} is uniformly bounded in L∞ ([0, T ], BV (R)). While by (3.1), (3.2), and the first equation in (1.7), we find that {∂t ρ } is uniformly bounded in L∞ ([0, T ], W −1,∞ (R)) + L2 ([0, T ], H −1 (R)). Notice by the compact embedding theorem that BV (R) >→>→ Lp (R) for any p < ∞. Thus by the Lions–Aubin lemma and a proof similar to that of Lemma 3 in [34], we find that there exists a ρ ∈ L∞ ([0, T ], BV (R)) for any T < ∞ such that (3.29)
ρ → ρ
in C([0, T ], Lp (K))
for all
T 0} and supp ρ0 =: {x, ρ0 (x) < 0}. Then m = |ρ| for + − almost all (t, x) ∈ D = D ∪ D , where − D+ = {(t, Φ+ t (x)) : x ∈ supp ρ0 } ,
+ D− = {(t, Φ− t (x)) : x ∈ supp ρ0 },
with Φ± t (x) being defined by (3.31)
dΦ± t (x) = ±u(t, Φ± t (x)), dt ± Φt (x)|t=0 = x.
Proof. First by multiplying signρ on both sides of (1.7), we find (3.32)
∂t |ρ | + div(u sign(ρ )(S (ρ ) − S (0))) ≤ −|ρ |S (0) + |ρ |.
Then by (3.1), it is trivial to prove that (3.33)
sign(ρ )(S (ρ ) − S (0)) − ρ L1 (K) → 0,
S (0)ρ L1 (K) → 0
as → 0. Hence by summing up (3.11), (3.32), (3.33), and taking → 0 in (3.32), we find ∂t m + div(uρ) ≤ 0.
(3.34)
Now let w = ρ + m. By summing up (3.15) and (3.34), we find ∂t w + div(uw) ≤ 0.
(3.35) Denote w (t, x) = satisfies (3.36)
j (y)w(t, x R2
− y) dy, and by [22, Lemma 2.3], we find that w
∂t w + div(uw ) ≤ R (t, x),
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EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
where R (t, x) = div(uw ) − j ∗ div(uw), and R → 0 in L1loc (R+ × R2 ). Equation (3.36) directly implies that (3.37)
dw (t, Φ+ t (x)) + ≤ (ρw )(t, Φ+ t (x)) + R (t, Φt (x)). dt
On the other hand, by (3.3), we find that suppρ ⊂ {(t, x) : |x| ≤ r + M t} =: B. Thus by [21, Lemma 1] or [33], we find that (3.31) has a unique global solution such that (3.38)
C(T )−1 |x1 − x2 |e
4πt
+ e ≤ |Φ+ t (x1 ) − Φt (x2 )| ≤ C(T )|x1 − x2 |
−4πt
,
and because divu = ρ, by [10, equation (74)], we have (3.39)
ρL∞ t R L1 (BT ) → 0 R (t, Φ+ t (x))L1 (K) ≤ e
as → 0, where BT = B ∩ {(t, x) : t ≤ T }. By summing up (3.37), (3.39), and letting → 0 in (3.37), we get for almost all x ∈ R2 that there holds (3.40)
dw(t,Φ+ t (x)) ≤ (ρw)(t, Φ+ t (x)), dt + w(t, Φt (x))|t=0 = 0, x ∈ supp ρ− 0.
Then the Gronwall inequality implies that (3.41)
w(t, Φ+ t (x)) = 0,
x ∈ supp ρ− 0,
t ∈ R+ .
By (3.38) and (3.41), we get (3.42)
d(t, x) = −ρ(t, x)
for a.e. (t, x) ∈ D+ .
Similarly, by subtracting (3.15) from (3.34) and letting q = m − ρ, we have (3.43)
∂t q − div(uq) ≤ 0.
By the proof of (3.42), we have (3.44)
m(t, x) = ρ(t, x)
for a.e. (t, x) ∈ D− .
Combining (3.42) and (3.44), we complete the proof of the proposition. Appendix. The construction of the approximate solutions. Let S (ξ) = |ξ| ∗ j (ξ), where j (ξ) is the standard Friedrich mollifier with supp j (·) ⊂ B (0). In the following, we are going to prove the global existence of smooth solutions to the following equations: d ∂t ρ + div(uS (ρ)) = ρ, (t, x) ∈ (0, x∞) × R , d = 1, 2, 3, −1 u = ∇ ρ for d = 2, 3, u = −∞ ρ dy for d = 1, (A.1) ρ|t=0 = ρ0 .
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QIANG DU AND PING ZHANG
Theorem A.1. Let s > d/2 + 1, ρ0 ∈ H0s (Rd ). Equation (A.1) has a unique global solution (ρ, u) such that ρ, ∇u ∈ L∞ ([0, T ], H s (Rd )) ∩ L2 ([0, T ], H s+1 (Rd )) for any T < ∞. Furthermore, (A.2) (A.3)
ρ(t, ·)L1 ≤ ρ0 L1 , ρ(t, ·)L∞ ≤ ρ0 L∞ + . t ρ2 (t, x) dx + 2 |∇ρ|2 dx ds ≤ ρ20 dx + 2 t Rd
0
Rd
Rd
Rd
|ρ0 | dx.
Proof. Following the standard argument for a nonlinear parabolic equation, we can prove the local existence and uniqueness of solution (ρ, u) to (A.1) such that ρ, ∇u ∈ L∞ ([0, T ], H s (Rd )) ∩ L2 ([0, T ], H s+1 (Rd )) for some positive constant T. Now let T ∗ be the lifespan of the solution (ρ, u) to (A.1). Then for t < T ∗ , notice the classical convex inequality: signρρ ≤ |ρ|. By multiplying signρ on both sides of (A.1), we find ∂t |ρ| + div(u signρ(S (ρ) − S (0))) ≤ −|ρ|S (0) + |ρ|.
(A.4)
Integrating the above inequality over Rd , we get the first inequality of (A.2) for t < T ∗ . Next, multiplying pρp−1 on both sides of the first equation of (A.1) with p an even integer, we get ∂t ρp + div(uF (ρ)) = G (ρ) + pρp−1 ρ,
(A.5)
ρ with F (ρ) = 0 pS (ξ)ξ p−1 dξ and G (ρ) = ρF (ρ) − pρp S (ρ). By the definition of S (ρ), we have pρ 0 S (ξ)ξ p−1 dξ − (p − 1)ρp+1 − p ρ + pρp Rd ξj (ξ) dξ, ρ pρ 0 S (ξ)ξ p−1 dξ − pρp S (ρ), G (ρ) = − pρ 0 S (ξ)ξ p−1 dξ + (p − 1)ρp+1 + p ρ − pρp Rd ξj (ξ) dξ, which together with the simple inequalities that S (ξ)ξ p−1 dξ ≤ p , p|ρ|p ≤ (p − 1)|ρ|p+1 + |ρ|, p
ρ ≥ , |ρ| ≤ ρ ≤ −,
S (ρ) ≥ 0
0
gives us G (ρ) ≤ p |ρ|.
(A.6)
Combining (A.2) with (A.6), and integrating (A.5) over Rd , we find t ρp dx + p(p − 1) ρp−2 |∇ρ|2 dx ds d d 0 R R t p p ρ0 dx + |ρ| dx ds ≤ ρp0 dx + p t ≤
(A.7)
Rd
0
Rd
Rd
Rd
|ρ0 | dx.
In particular, by taking p = 2 in the above, we get (A.3). Moreover, for any compact subset K of Rd , (A.7) implies that (A.8)
K
p
ρ (t, x) dx
p1
1
1
≤ ρ(t, ·)Lp ≤ ρ0 Lp + t p ρ0 Lp 1 .
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1297
Letting p → ∞ in (A.8), we prove the second inequality of (A.2) for t < T ∗ . Now let E(t, x) be the fundamental solution of the heat operator (∂t − ); then d
|x|2
E(t, x) = (4πt)− 2 e− 4t ,
(A.9)
t > 0,
by (A.1), and ρ can also be written by the following form: ρ(t, x) = −
t Rd
0
E(t − s, x − y)div(uS (ρ))(s, y) dy ds +
Rd
E(t, x − y)ρ0 (y) dy
for t < T ∗ . By integrating by parts in the above formula, we find t (A.10) ρ(t, x) = − ∇x E(t − s, x − y)(uS (ρ))(s, y) dy ds + ρ0 (t, x), Rd
0
and consequently, ∂x ρ(t, x) = −
t Rd
0
(A.11)
∇x E(t − s, x − y)(∂x uS (ρ) + uS (ρ)∂x ρ)(s, y) dy ds + ∂x ρ0 (t, x).
Thus by the Hausdorff–Young inequality and the fact that |S (ξ)| ≤ 1, we get ∂x ρ(t, ·)L1 ≤ C(sup ρ(t, ·)L1 + uL∞ ) t>0
(A.12) ∂x ρ(t, ·)L∞
0
t
1
(t − s)− 2 (∂x u(s, ·)L∞
+ ∂x ρ(s, ·)L1 ) ds + ∂x ρ0 L1 , t 1 ≤ C(ρ(t, ·)L∞ + uL∞ ) (t − s)− 2 (∂x u(s, ·)L∞ 0
(A.13)
+ ∂x ρ(s, ·)L∞ ) ds + ∂x ρ0 L∞ .
On the other hand, for d = 2 and 3, by the second equation of (A.1), we have x−y |u(t, x)| = ρ(t, y) dy d Rd |x − y| 1 ρ(t, ·)L1 ≤ ρ(t, ·)L∞ dy + d−1 Rd−1 |x−y|≤R |x − y| (A.14)
≤ CRρ(t, ·)L∞ + ρ(t,·)
ρ(t, ·)L1 . Rd−1
1
1 Taking R = ( ρ(t,·)LL∞ ) d in (A.14), we find 1
1− 1
(A.15) u(t, ·)L∞ ≤ Cρ(t, ·)Ld 1 ρ(t, ·)L∞d ≤ C (ρ(t, ·)L1 + ρ(t, ·)L∞ ) , while for d = 1, the second equation of (A.1) directly implies that (A.16)
u(t, ·)L∞ ≤ ρ(t, ·)L1 .
Similar to the proof of (A.15) and (A.16), we have (A.17)
∂x u(t, ·)L∞ ≤ C (∂x ρ(t, ·)L1 + ∂x ρ(t, ·)L∞ ) .
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QIANG DU AND PING ZHANG
Now let us set y(t) = ∂x ρ(t, ·)L1 + ∂x ρ(t, ·)L∞ . By combining (A.2) with (A.12)– (A.17), we find y(t) ≤ C
0
t
1
(t − s)− 2 y(s) ds + y0 ,
t < T ∗.
The Gronwall inequality yields that y(t) ≤ y0 eC
(A.18)
√
t
,
t < T ∗.
On the other hand, standard energy estimates (see [24]) show that if T ∗ < ∞, then (A.19)
lim (∂x u(t, ·)L∞ + ∂x ρ(t, ·)L∞ ) = ∞.
t→T ∗
This contradicts (A.17) and (A.18). Thus T ∗ = ∞. This completes the proof of the theorem. Acknowledgments. The authors would like to thank the referees for their valuable suggestions. REFERENCES [1] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interaction, Comm. Pure Appl. Math., 55 (2002), pp. 30–70. [2] S. J. Chapman, A mean-field model of superconducting vortices in three dimensions, SIAM J. Appl. Math., 55 (1995), pp. 1259–1274. [3] S. J. Chapman, A hierarchy of models for type-II superconductors, SIAM Rev., 42 (2000), pp. 555–598. [4] S. J. Chapman, J. Rubinstein, and M. Schatzman, A mean-field model of super-conducting vortices, European J. Appl. Math., 7 (1996), pp. 97–111. [5] Z. Chen and Q. Du, A non-conforming finite element methods for a mean field model of superconducting vortices, Math. Model. Numer. Anal., 34 (2000), pp. 687–706. [6] M. G. Crandall, H. Ishi, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), pp. 1–67. [7] J. M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 14 (1991), pp. 553–586. [8] R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), pp. 667–689. [9] R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math., 40 (1987), pp. 301–345. [10] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), pp. 511–547. [11] A. T. Dorsey, Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach, Phys. Rev. B, 46 (1992), pp. 8376–8392. [12] Q. Du, Convergence analysis of a numerical method for a mean field model of superconducting vortices, SIAM J. Numer. Anal., 37 (2000), pp. 911–926. [13] Q. Du, M. Gunzburger, and H. Lee, Analysis and computation of a mean field model for superconductivity, Numer. Math., 81 (1999), pp. 539–560. [14] Q. Du, M. D. Gunzburger, and J. S. Peterson, Analysis and approximation of the Ginzburg– Landau model of superconductivity, SIAM Rev., 34 (1992), pp. 54–81. [15] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), pp. 383–404. [16] W. E, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Phys. Rev. B, 50 (1994), pp. 1126–1135. [17] W. E, private communication, 1998. [18] C. Elliott and V. Styles, Numerical analysis of a mean field model of superconductivity, IMA J. Numer. Anal., 21 (2001), pp. 1–51.
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¨tzle, and B. E. E. Stoth, Viscosity solutions of a degenerate [19] C. M. Elliott, R. Scha parabolic-elliptic system arising in the mean-field theory of super-conductivity, Arch. Ration. Mech. Anal., 145 (1998), pp. 99–127. [20] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. Math. 74, AMS, Providence, RI, 1990. [21] F. Lin and P. Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices, Discrete Contin. Dynam. Systems, 6 (2000), pp. 121–142. [22] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Oxford Lecture Ser. Math. Appl. 3, Clarendon Press, Oxford, 1996. [23] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible Models, Oxford Lecture Ser. Math. Appl. 10, Clarendon Press, Oxford, 1998. [24] A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. [25] A. J. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., 39 (1986), pp. 187–220. [26] A. J. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J., 42 (1993), pp. 921–939. [27] A. Majda, G. Majda, and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case, Phys. D, 74 (1994), pp. 268–300. [28] A. Majda, G. Majda, and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case, Phys. D, 79 (1994), pp. 41–76. [29] G. Richardson and B. Stoth, Ill-posedness of the mean-field model of superconducting vortices and a possible regularisation, European J. Appl. Math., 11 (2000), pp. 137–152. [30] R. Schatzle and V. Styles, Analysis of a mean field model of superconducting vortices, European J. Appl. Math., 10 (1999), pp. 319–352. [31] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. [32] M. E. Taylor, Pseudo-Differential Operators and Nonlinear PDE, Birkh¨ auser-Verlag, Boston, 1991. [33] V. I. Yudovich, Nonstationary flow of an ideal incompressible liquid, USSR Comput. Math. Phys., 3 (1963), pp. 1407–1456. [34] P. Zhang and Y. Zheng, Rarefactive solutions to a nonlinear variational wave equation of liquid crystals, Comm. Partial Differential Equations, 26 (2001), pp. 381–419.
c 2003 Society for Industrial and Applied Mathematics
EXISTENCE OF WEAK SOLUTIONS TO SOME VORTEX DENSITY MODELS∗ QIANG DU† AND PING ZHANG‡ Abstract. We study the weak solutions to equations arising in the modeling of vortex motions in superfluids such as superconductors. The global existence of measure-valued solutions is established with a bounded Radon measure as initial data. Moreover, we get a local space-time Lq estimate for the continuous part of the solution, and we prove the global existence of a distributional weak solution for a particular case. We also consider a modification to the model in order to physically account for the different signs of vortices, and we present, in one space dimension, the global existence of weak solutions with the initial data in BV for the modified model. Key words. quantized vortices, vortex density, hydrodynamics, vortex sheets, weak convergence, measure-valued solutions AMS subject classifications. 35Q55, 35B, 35K PII. S0036141002408009
1. Introduction. In this paper, we study the concentration phenomenon of the approximate solution sequences to the equations ∂t ρ + div(uρ) = 0, (t, x) ∈ (0, ∞) × R2 , u = M ∇−1 ρ, (1.1) ρ|t=0 = ρ0 , with ρ0 being a bounded Radon measure and M being a constant orthogonal matrix of the form cosθ −sin θ M (θ) = . sin θ cos θ Our investigation yields the global existence of a measure-valued solution to (1.1) and the classical weak solution to (1.1) if ρ0 is a bounded positive (resp., negative) Radon measure when cos θ > 0 (resp., cos θ < 0). For the case cos θ = 1, our results here extend those available in the literature (see, for instance, [21]). In the more general case, our study is related to the mathematical study of incompressible fluids as well as the vortex state in superfluids. Indeed, when cos θ = 0, (1.1) is the classical two-dimensional incompressible Euler equations, which can be rewritten in the velocity formulation ∂t u + div(u ⊗ u) = −∇P, (t, x) ∈ (0, ∞) × R2 , divu = 0, (1.2) u|t=0 = u0 . ∗ Received by the editors May 21, 2002; accepted for publication (in revised form) November 18, 2002; published electronically May 12, 2003. This work was supported in part by the Chinese NSF, innovation grants from the Chinese Academy of Sciences, the state key basic research project G199903280, and the U.S. NSF grant DMS-0196522. This work was completed during Ping Zhang’s visit to Penn State University. http://www.siam.org/journals/sima/34-6/40800.html † Department of Mathematics, Penn State University, University Park, PA 16802, and Lab for Scientific and Engineering Computing, CAS, China ([email protected]). ‡ Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China ([email protected]).
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The initial value problem to (1.2) with u0 = (u10 , u20 ) ∈ L2loc (R2 ) and ω0 = ∂2 u10 − ∂1 u20 ∈ M(R2 ) is an outstanding open problem, known as the vortex sheets problem, in incompressible fluid mechanics (see [25]). In 1991, Delort [7] proved the global existence of weak solutions to this problem when ω0 is a bounded Radon measure without negative singular part. Later, Majda [26] obtained the same result by the vanishing viscosity limit to the two-dimensional incompressible Navier–Stokes equations. When ω0 is a general Radon measure, only a measure-valued solution seems possible for this problem; see [8, 9, 27, 28] for more details. In recent years, studies of the stability, dynamics, and interactions of the vortices in both classical fluids and superfluids have received a lot of attention. For example, in the mezoscale Ginzburg–Landau models of superconductors [14], the individual vortices are resolved and their interactions and dynamics are studied in great detail. On a macroscopic level, when the number of vortices becomes exceedingly large, it is advantageous to model the vortex state using a vortex density function [4, 16]. When cos θ = 1, (1.1) has been obtained as the hydrodynamic limit of Ginzburg–Landau vortices governing by gradient dynamics. A formal derivation was given in [16], and a rigorous justification was given in [21]. Studies in this direction also include [2, 4, 15] on model derivation, [19, 29, 30] on mathematical analysis, and [5, 12, 13, 18] on numerical simulations (see [3] for additional references). The general case of θ = 0 corresponds to a complex time relaxation in the gradient dynamics. In [21], the global existence of weak solutions to (1.1) with cos(θ) = 1 and ρ0 being a positive bounded Radon measure was established. In the case of ρ0 taking on different signs, the notion of weak solutions to (1.1) with cos(θ) = 0 requires further study, as additional difficulties do arise. A similar but somewhat modified version of (1.1) was studied in [2, 4] when the vortices are of different signs. Taking the London approximation to the induced magnetic field into account, a system of equations similar to (1.1) with cos(θ) = 1 was derived in [4]: ∂t ρ + div(u|ρ|) = 0, (t, x) ∈ (0, ∞) × R2 , u = ∇(λ2 − I)−1 ρ, (1.3) ρ|t=0 = ρ0 . Here, λ denotes the penetration depth. The density function ρ is allowed to change sign in order to represent vortices of different signs. The general case of θ = 0 can also be easily derived when the time relaxation parameter becomes complex valued [17]. In [11], such an approach was taken to account for the Hall effect. A vector-valued version of (1.3) was also available [4] to account for the three-dimensional effect. The existence and uniqueness of a viscosity solution to an equation similar to (1.3) was proved in [19] by the viscosity solution method in [6] for an R2 -valued function ρ, since a scalar stream function ψ can be found in that case such that ρ = ∇⊥ ψ. Such a technique obviously is not applicable here. To our knowledge, the general existence to (1.3) is still open except for the stationary solutions studied in [4]. To draw an analogy with (1.1), we consider a modification of the above equation: ∂t ρ + div(u|ρ|) = 0, (t, x) ∈ (0, ∞) × R2 , u = ∇−1 ρ, (1.4) ρ|t=0 = ρ0 . Notice that when ρ0 ≥ 0, (1.4) is the same as (1.1).
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
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This paper consists of two main parts. In the first part, we study the global existence of weak solutions to (1.1) with general Radon measure as initial data under the condition that cos θ = 0 and obtain more general results than those given in [21]. Without loss of generality, we restrict ourselves to the case cos θ > 0; the results for cos θ < 0 can be similarly obtained. In the second part of the paper, we present an existence result to (1.4) in one space dimension. To introduce our main results, let us examine the general procedure on proving the global existence of weak solutions. As in [9], the first step is to construct the approximate solution sequences. For simplicity, let us define the following cut-off function: ξ, ξ ≥ −1/, T (ξ) := (1.5) −1/, ξ ≤ −1/. We study the approximate solution sequence to (1.1) constructed by the equations ∂t ρ + u ∇ρ = −cos θT (ρ )ρ , (t, x) ∈ R+ × R2 , u = M (θ)∇−1 ρ , (1.6) ρ |t=0 = ρ0, , 1, |x| ≤ 1, where ρ0, = (ρ0 χ ) ∗ j , χ (x) = χ( x ), χ ∈ Cc∞ (R2 ), χ(x) = 0, |x| ≥ 2, and j (x)
is the standard Friedrich’s mollifier with suppj ⊂ B (0). Let S (ξ) = |ξ| ∗ j . The approximate solution sequence to (1.4) may be defined by the following equation: ∂t ρ + div(u S (ρ )) = ρ , (t, x) ∈ R+ × R2 , u = ∇−1 ρ , (1.7) ρ |t=0 = (ρ0 χ ) ∗ j . Now, a main result of this paper can be stated in the following theorem. Theorem 1.1. Let ρ0 ∈ M(R2 ) and cosθ > 0. Then there exist a subsequence of {ρ , u } constructed by (1.6) (still denoted by {ρ , u } for convenience), functions 1,q ρ ∈ Lqloc (R+ × R2 ) ∩ L∞ (R+ , L1 (R2 )) and u ∈ Lqloc (R+ , Wloc (R2 )) for any q < 2, and + + 2 a positive Radon measure µ ∈ M (R × R ) such that the following hold: 1. The following convergence properties and estimates hold: (1.8) (1.9) (1.10) (1.11)
ρ ρ weakly in Lqloc (R+ × R2 ), 1,q u u weakly in Lqloc (R+ , Wloc (R2 )), 1 ρ 1ρ ≤− 1 µ in the sense of ρ + ∞ dµ ≤ | dρ0 (x)|. 0
R2
M(R+ × R2 ),
R2
2. The following decay estimates hold: ρ(t, x) ≤
(1.12)
cos θ t
for a.e.
(t, x) ∈ R+ × R2 .
3. For all test functions ϕ ∈ Cc∞ ([0, ∞) × R2 ), there holds ∞ (1.13) (ρ∂t ϕ + ρu∇ϕ + µϕ) dx dt + ϕ(0, x)ρ0 dx = 0 0
R2
R2
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QIANG DU AND PING ZHANG
and u = M (θ)∇−1 ρ.
(1.14)
Definition 1.1. We call (ρ, u, µ) the measure-valued solution to (1.1) if (ρ, u, µ) satisfies (1.13) and (1.14). A similar definition was used by Alexandre and Villani in the study of the Boltzmann equation without Grad’s angular assumption to the cross section [1]. There, f (t, x, v) was defined as a renormalized solution to the Boltzmann equation with a defect measure µ(t, x, v) if, for all nonlinearity β ∈ C 2 (R+ , R+ ) satisfying β(0) = 0, C 0 < β (f ) ≤ 1+f , β (f ) < 0, there holds ∂t β(f ) + v · ∇β(f ) = β (f )Q(f, f ) + µ in the sense of distributions. One may check [1] for more details. When ρ0 ∈ M+ (R2 ), we have the following improvement of Theorem 1.1. Corollary 1.1. Let 0 < cos θ < 1 and ρ0 ∈ M+ (R2 ). Then µ = 0 in (1.13), and ρ ∈ L2 ((0, ∞) × R2 ). Moreover, (1.15)
ρL2 ((0,∞)×R2 ) ≤ Cρ0 (R2 ).
If 0 ≤ ρ0 ∈ Lp (R2 ) for 1 < p < ∞, we have the following better estimate for ρ: T ρp (T, x) dx+(p−1)cos θ ρp+1 (t, x) dx dt ≤ ρp0 (x) dx for a.e. T ∈ R+ . R2
(1.16)
0
R2
R2
Remark 1.1. In comparison with the measure-valued solutions to (1.2) in [8] and that of the one-dimensional two-component Vlasov–Poisson equations in [27, 28], the measure-valued solution to (1.1) here is much more explicit and closer to the distributional weak solution of (1.1). By (1.10), if ρ0 is a sign-changing Radon measure, µ may not be 0 even if {ρ } strongly converges to ρ in Lqloc (R+ × R2 ) for any q < 2. Moreover, from the corollary, we can see that if cos θ > 0 and ρ0 is a positive Radon measure, then the approximate solutions satisfy ρ (t, x) ≥ 0 for all (t, x) ∈ R+ × R2 , which in turn implies, by the definition of µ, that µ = 0. Thus, (ρ, u) is the classical distributional weak solution to (1.1). 2 Remark 1.2. If cos θ > 0 and 0 ≤ ρ0 (x) ∈ L∞ comp (R ), following exactly the same procedure as that in [21] and [33], we can prove the uniqueness of the weak solution in the above corollary. Remark 1.3. We can replace the second equation in (1.1) by u = −M (θ)∇(−λ2 + −1 1) ρ in correspondence to the equations derived in [4]. Step-by-step modifications of the proofs given here will yield similar results for such equations as those in Theorem 1.1. In one space dimension, (1.4) takes on the following form: ∂t ρ + ∂x (u|ρ|) = 0, (t, x) ∈ (0, ∞) × R, x u = −∞ ρ(t, y) dy, (1.17) ρ|t=0 = ρ0 . Then we have the following existence result to (1.17). Theorem 1.2. Let ρ0 ∈ BV (R). Then the weak limit (ρ, u) obtained by the vanishing viscosity of (1.7) satisfies (1.17) in the sense of distributions, and
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
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ρ ∈ C([0, T ], Lp (K)) ∩ L∞ ([0, T ], BV (R)), u ∈ L∞ ([0, T ], W 1,p (R)) for any T < ∞, 2 < p < ∞, and any compact subset K of R. Furthermore, (1.18) |ρ(t, x)| dx ≤ |ρ0 | dx, | dρ(t, x)| ≤ | dρ0 |,
R
R
R
R
where R | dρ| denotes the total variation of ρ, if supp ρ0 ⊂ Br (0), supp ρ(t, ·) ⊂ {x : |x| ≤ r + M t} with M the L∞ bound of u. Remark 1.4. (1) An explicit solution to (1.17) may be constructed: let ρ0 be defined by ρ0 (x) = 1 in (0, 1), ρ0 (x) = −1 in (1, 2), and ρ0 (x) = 0 elsewhere; then a global weak solution to (1.17) may be defined by 1 x ∈ (0, 1), 1+t , −1 x ∈ (1, 2), ρ(t, x) = 1+t , 0 otherwise. Here, x = 1 is the shock line of this solution. Naturally, this leads to a conjecture that shocks will be formed for d = 2 when ρ0 changes sign. In general we cannot prove |ρ | m = |ρ|. Nevertheless, we can prove that ∂t ρ + div(um) = 0,
u = ∇−1 ρ
holds in the sense of distributions and m = |ρ| for almost all (t, x) in a subset of R+ × R2 ; see Proposition 3.1 for more details. (2) With d = 2, to get the uniform L∞ ([0, T ], L1 (R2 )) estimate for {∂x ρ }, the solution sequence to (1.7), we need the uniform L∞ estimate for ∇u (see the proof of Lemma 3.2 for details). But with ρ being uniformly bounded, we cannot get the desired estimate for ∇ ⊗ u = ∇ ⊗ ∇−1 ρ by the singular integral operator theory [31]. However, in the case of d = 1, by the second equation of (1.17), we find ∂x u = ρ , which gives the desired estimate for ∂x u . Remark 1.5. Again, one may replace the second equation of (1.17) by u = −∂x (−λ2 ∂xx + 1)−1 ρ and, using the same type of arguments as those in Theorem 1.2, prove similar results. We now introduce some notation that will be used throughout the paper. We let Br (0) = {x : |x| ≤ r} and denote by M(Ω) the bounded Radon measure space on Ω, by M+ (Ω) the bounded positive Radon measure space on Ω, and BV (R) = {f : f ∈ L1 (R), ∂x f ∈ M(R)}. We use C(a, b, . . .) as a uniform constant which only depends on the listed variables and may change from line to line. The proofs of the above theorems are given in later sections. 2. Proof of Theorem 1.1. Now let us first prove the global existence of solutions to (1.6). For convenience, we omit the subscript in the approximate solution sequence {(ρ , u )} in the following lemma. Lemma 2.1 (solution of (1.6) with smooth data). For ρ0 ∈ Cc∞ (R2 ), (1.6) has a global strong solution (ρ, u) such that ρ ∈ L∞ ([0, T ], W 1,p (R2 )), ∇u ∈ L∞ ([0, T ], W 1,p (R2 )) for any p > 1, T < ∞, and (2.1)
ρ(t, ·)L1 ≤ ρ0 L1
and
ρ(t, x) ≤
cos θ , t
t > 0.
Proof. 1. (Blow-up principle.) Following the standard argument for a nonlinear hyperbolic equation, we can prove the local existence of solution (ρ, u) to (1.6) with
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QIANG DU AND PING ZHANG
smooth data such that ρ, ∇u ∈ L∞ ([0, T ], W 1,p (R2 )) for some positive constant T and any p < ∞. Now, let T ∗ be the lifespan of the solution (ρ, u) to (1.6). We are going to show that if T ∗ < ∞, (2.2)
lim ρ(t, ·)L∞ = ∞.
t→T ∗
In fact, for any even positive number p, it follows from (1.6) that (2.3)
∂t (∂xi ρ)p + u∇(∂xi ρ)p + p∂xi u∇ρ(∂xi ρ)p−1 = −pcos θ∂xi (T (ρ)ρ)(∂xi ρ)p−1 .
Noticing that divu = cos θρ, |∂xi (T (ρ)ρ)| ≤ 2|ρ||∂xi ρ|, integrating the above equation over R2 , and using integration by parts, we get d (2.4) |∂xi ρ|p dx ≤ ((2p + 1)cos θρL∞ + p∂xi uL∞ ) |∇ρ|p dx. dt R2 R2 Let us take χ(ξ) ∈ Cc∞ (R2 ), χ(D) the corresponding pseudodifferential operator with symbol χ(ξ); then by singular integrals theory [31, 32], we have (2.5)
χ(D)∇ ⊗ ∇−1 ρL∞ ≤ C∇ ⊗ ∇−1 ρLp ≤ CρLp .
While by Lemma B.1.C of [32], for all p > 2, we find ρW 1,p −1 (1 − χ(D))∇ ⊗ ∇ ρL∞ ≤ CρL∞ 1 + log (2.6) . ρL∞ Summing up the second equation of (1.6) and inequalities (2.5) and (2.6), we find ρW 1,p ∇uL∞ ≤ C ρL∞ 1 + log (2.7) + ρLp . ρL∞ A simple interpolation result gives us (2.8)
p−1
1
p ρLp ≤ ρL∞ ρLp 1 .
Summing up (2.4) and (2.7)–(2.8) we obtain 1 d ∇ρLp p ∇ρ(t, ·)Lp ≤ C ρL∞ 1 + log + ρL1 ∇ρpLp . (2.9) 2 dt ρL∞ Then the Gronwall inequality yields that (2.10)
∇ρ(t, ·)Lp ≤ C(T, ρL1 , ρL∞ )∇ρ0 Lp .
On the other hand, by multiplying signρ on both sides of (1.6), we find by (1.5) that (2.11)
∂t |ρ| + div(u|ρ|) = cos θ(ρ − T (ρ))|ρ| ≤ 0.
Integrating (2.11) over R2 , we get the first inequality of (2.1). Summing up (2.1) and (2.10), we complete the proof of the claim (2.2). 2. (Estimate of ρL∞ .) By (2.7) and the classical theory on ordinary differential equations, the equation dΦt (x) = u(t, Φt (x)), dt (2.12) Φt (x)|t=0 = x
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1285
has a unique solution Φt (x) ∈ C([0, T ∗ ) × R2 ), and ∂x Φt (x) ∈ L∞ ([0, T ] × R2 ) for any T < T ∗ . Then, by the first equation of (1.6), we have dρ(t, Φt (x)) ≤ 0, dt which implies that ρ(t, ·) ≤ ρ0 L∞ .
(2.13)
This together with (1.5) shows that (2.14)
T (ρ)L∞ ≤ max
1 , ρ0 L∞
.
Thus by the first equation of (1.6) and by (2.12), we have d|ρ|(t, Φt (x)) 1 ∞ |ρ|(t, Φt (x)), ≤ cos θ max , ρ0 L dt which together with the Gronwall inequality yields that 1 , ρ0 L∞ t ρ0 L∞ . (2.15) ρL∞ ≤ exp cos θ max Summing up (2.2) and (2.15), we get the global existence of strong solutions to (1.6) with smooth initial data. 3. (Decay estimate.) By the first equation of (1.6) and by (2.12), we have (2.16)
dρ(t, Φt (x)) = −cos θ(T (ρ)ρ)(t, Φt (x)), dt
which implies that if ρ0 (x) ≤ 0, then ρ(t, Φt (x)) ≤ 0, and if ρ0 (x) ≥ 0, then ρ(t, Φt (x)) ≥ 0. So to prove the one-sided decay estimate (2.1), we need to consider only the points where ρ0 (x) > 0. By (1.5), T (ρ(t, Φt (x)) = ρ(t, Φt (x)). Solving (2.16), we get (2.17)
ρ(t, Φt (x)) =
cos θ cos θρ0 (x) < , 1 + tρ0 (x) t
t > 0.
Summing up the above, we get the second inequality of (2.1). This completes the proof of the lemma. Next let us get the key uniform space-time estimate for the approximate solution sequence {ρ } constructed in Lemma 2.1. Lemma 2.2 (L1+α estimate). Let ρ0, ∈ L1 (R2 ), α ∈ (0, 1), T, R > 0. Then for the solutions (ρ , u )>0 of (1.6), there holds the estimate T |ρ |1+α dx dt ≤ Cα,T,R , (2.18) 0
|x|≤R
where the constant Cα,T,R depends only on the L1 norm of ρ0, and the listed variables. Proof. 1. (Elementary estimate.) We first assume α = d2 /d1 ∈ (0, 1/2), where d1 and d2 are odd positive integers. Let χ ∈ Cc∞ (R2 ), χ ≥ 0 and χ = 1 on {x||x| ≤ R},
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QIANG DU AND PING ZHANG
ξ with supp χ ⊂ {x||x| ≤ R + 1}. Set η(ξ) = α 0 max(1, |s|)α−1 ds for ξ ∈ R1 such that η (ξ) = α max(1, |ξ|)α−1 . Multiplying (1.6) by χη (ρ ), integrating the resulting identity over [0, T ] × R2 , and performing integration by parts several times, we obtain T (2.19) cos θ χ(ρ η(ρ ) − ρ T (ρ )η (ρ )) dx dt R2
0
=
R2
χη(ρ ) dx|T0
−
0
T
R2
∇χu η(ρ ) dx ds.
First, by the second equation of (1.6), for all test functions φ(x), ψ(x) ∈ Cc∞ (R2 ), we have x−y (ψ(x) − ψ(y)) ρ (t, y) dy φ(x)ψ(x)u (t, x) = M (θ)φ(x) |x − y|2 2 R x−y φ(x)ψ(y) ρ (t, y) dy. (2.20) + M (θ) |x − y|2 R2 x−y x−y ∞ Notice that φ(x)ψ(y) |x−y| 2 = φ(x)ψ(y)ζ(|x − y|) |x−y|2 , where ζ(z) ∈ Cc (R) with ζ(z) = 1 for z ∈ suppφ + supp ψ. And trivially ζ(z) |z|z 2 ∈ Lp (R2 ) for all p < 2, by the Hausdorff–Young inequality to the second term of (2.20), we get
φψu (t, ·)Lp ≤ (sup |∇ψ|φLp + cφ,ψ )ρ (t, ·)L1
for all 1 ≤ p < 2.
In particular, by taking φ(x) = ψ(x) = 1 for |x| ≤ R + 1, we get
p1 (2.21)
|x|≤R+1
|u (t, x)|p dx
≤ CR ρ0, L1
for all 1 < p < 2.
Note that α < 12 , and thus by (2.1) and (2.21), we have
1−α
α T T 1 ∇χu η(ρ ) dx ds ≤ |u | 1−α dx |ρ | dx dt 0 R2 0 |x|≤R+1 |x|≤R+1 ≤ C1 (R, ρ0, L1 ) .
(2.22)
By (2.1) and the definition of η, we have χη(ρ )(T, x) dx 2 R ≤ χ(α + |ρ |α )(T, x) dx + α (2.23)
|ρ |≥1
2
≤ 2πα(R + 1) +
|x|≤R+1
|ρ |≤1
χ(x)|ρ | dx
α
|ρ |(T, x) dx
(2π(R + 1)2 )1−α + α2π(R + 1)2
≤ C2 (α, R, ρ0, L1 ). Finally it follows from the definition of α and η that T χ(x)(ρ η(ρ ) − ρ T (ρ )η (ρ )) dx dt 0
(2.24)
=
0
≥
0
R2 T
T
R2
R2
1|ρ |≥1 χ (1 − α)ρ1+α + α(ρ1+α − T (ρ )ρα ) + αρ dx 1|ρ |≥1 χ((1 − α)ρ1+α + αρ ) dx,
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1287
where 1|ρ |≥1 is the characteristic function on the set {(t, x) : |ρ (t, x)| ≥ 1}. Summing up (2.19)–(2.24), we find (2.25)
T
0
χρ1+α
|ρ |≥1
1 dx dt ≤ 1−α
α
R2
χ|ρ | dx + C1 + C2
for all α = d2 /d1 ∈ (0, 1/2). 2. (Inductive step 1.) Next, we are going to show by the bootstrap method that (2.25) holds for all α ∈ (0, 1). First, let us take α = d2 /d1 ∈ (0, 5/6) with d1 , d2 positive odd integers. In particular, due to the arbitrariness of R, by interpolation, and by (2.25), we have (2.26)
T
0
|x|≤R+1
|ρ |p1 dx dt ≤ C(R, T, ρ0, )
for all p1
1, again by [20] and (2.37), we have
ρp (T, x) dx + (p − 1)cos θ 2 R ≤ lim→0 ≤ lim→0
(2.56)
R2
R2
T
0
R2
ρp+1 dx dt
ρp (T, x) dx
ρp0, (x) dx =
+ (p − 1)cos θ
R2
T
0
R2
ρp+1
dx dt
ρp0 dx.
Then, by summing (2.55) and (2.56), we complete the proof of Corollary 1.1. 3. Proof of Theorem 1.2 and the remarks. Again the first step in the proof of Theorem 1.2 is to construct the approximate solution sequence {(ρ , u )}. By Theorem A.1, we immediately have the following lemma. Lemma 3.1 (solution of (1.7) with smooth data). Let d = 1, 2, ρ0 ∈ L∞ (Rd ). Then (1.7) has a global smooth solution (ρ , u ) such that ρ , ∇u ∈ L∞ ([0, T ], H s (Rd ))∩ L2 ([0, T ], H s+1 (Rd )) for any s > d2 + 1, T < ∞, and (3.1) (3.2)
ρ (t, ·)L1 ≤ ρ0 L1 , ρ (t, ·)L∞ ≤ ρ0 L∞ + , t 2 2 2 2 ρ (t, x) dx + 2 |∇ρ | dx ds ≤ ρ0 dx + t Rd
0
Rd
Rd
Rd
1
|ρ0 | dx. 1− 1
Furthermore, if supp ρ0 ⊂ Br (0), we denote M = (ρ0 L∞ + 1) d ρ0 L1 d ; then for r > and (t, x) ∈ Ωo =: {(t, x) : |x| ≥ r + M t, t ≥ 0}, there holds (3.3)
|ρ (t, x)| ≤ ρ0 L∞ exp[−1 (r + M t − |x|) + tρL∞ ] ≡ Q (t, x).
Proof. First by the third equation of (1.7), the global existence and uniqueness of solution to (1.7) is a direct consequence of Theorem A.1. Moreover, (A.2) and (A.3) imply (3.1) and (3.2), respectively. Consequently, (A.15) implies that u L∞ ≤ M. Next, we rewrite the first equation of (1.7) as (3.4)
∂t ρ + u S (ρ )∇ρ + S (ρ )ρ = ρ .
Let us denote L = ∂t + u S (ρ )∇ + S (ρ ) − . Notice by the definition of S (ξ) that |S (ρ )| ≤ 1 and S (ρ )L∞ ≤ ρ L∞ . We find (3.5) (3.6)
LQ ≥ 0 on Ωo , Q ||x|=r+M t ≥ ρ0 L∞ ,
Q |t=0,|x|≥r ≥ 0.
Hence by the maximum principle, we have (3.7)
ρ (t, x) ≤ Q (t, x)
for all (t, x) ∈ Ωo .
Similarly, we can prove (3.8)
−ρ (t, x) ≤ Q (t, x)
for all (t, x) ∈ Ωo .
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QIANG DU AND PING ZHANG
Combining (3.7) with (3.8), we get (3.3). This completes the proof of the lemma. With this lemma, by (1.7), (3.1), (3.2), and [31], we have that ∂t u = ∂t ∇−1 ρ = −∇−1 div(u S (ρ )) + ∇ρ
(3.9)
is uniformly bounded in L∞ ([0, T ], L2 (Rd )). While again by (1.7), (3.1), and [31], we have is uniformly bounded in L∞ ([0, T ], W 1,p (Rd ))
u
(3.10)
for any 2 < p < ∞. Thus by combining (3.9), (3.10), the Lions–Aubin lemma, and the similar proof of Lemma 3 in [34], we obtain that there exists a subsequence of {u }, which we still denote by {u }, and some u ∈ L∞ ([0, T ], W 1,p (Rd )) such that u → u
(3.11)
uniformly on any compact subset of [0, T ] × Rd .
Trivially by (3.1), there exist ρ, m ∈ L∞ ([0, T ] × Rd ) such that ρ ρ weakly ∗ in L∞ ([0, T ] × Rd ), |ρ | m weakly ∗ in L∞ ([0, T ] × Rd ).
(3.12) (3.13)
While by the definition of S (ξ), for any compact subset K of [0, T ] × R2 , we have S (ρ ) − |ρ |L1 (K) → 0
(3.14)
as
→ 0.
Thus by combining the first equation of (1.7) with (3.11)–(3.14), we have that ∂t ρ + div(um) = 0
(3.15)
holds in the sense of distributions. Summing up the second equation of (1.7) and (3.11) and (3.12), we get u = ∇−1 ρ .
(3.16)
Thus, in order to prove that (ρ, u) is indeed a global weak solution to (1.4), we only need to prove that m = |ρ|. However, only in one space dimension, and ρ0 ∈ BV (R), we can prove that d(t, x) = |ρ|(t, x) for almost all (t, x) ∈ R+ × R. In order to do so, let us first present the following lemma. Lemma 3.2. Let ρ0 ∈ BV (R). Then (3.17) |∂x ρ (T, x)| dx ≤ e3T | dρ0 |,
R
R
where R | dρ0 | is the total variation of ρ0 . Proof. Let g be the solution of the adjoint equation (3.18)
∂t g + u S (ρ )∂x g − (S (ρ ) + ρ S (ρ ))g + ∂xx g = 0,
with the Cauchy data g(T, ·) = γ ∈ C0∞ ({x : |x| < R}), and γL∞ ≤ 1. Let τ = T −t, h = e−3τ g; then by (3.18), we have (3.19)
∂τ h − u S (ρ )∂x h + (S (ρ ) + ρ S (ρ ) + 3)h − ∂xx h = 0.
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1293
We assume that h reaches its minimum value at (τ0 , x0 ). Then we claim that either h(τ0 , x0 ) ≥ 0
(3.20)
or τ0 = 0.
Otherwise, if h(τ0 , x0 ) < 0 and τ0 > 0, we have by the definition of (τ0 , x0 ) that (3.21)
∂t h(τ0 , x0 ) = 0,
∂x h(τ0 , x0 ) = 0,
∂xx h(τ0 , x0 ) ≥ 0,
which implies that {∂τ h − u S (ρ )∂x h + (S (ρ ) + ρ S (ρ ) + 3)h − ∂xx h} (t0 , x0 ) < 0 as S (ρ ) ≥ 0, ρ S (ρ )+3 > 0. This contradicts (3.19), which proves the claim (3.20). Hence h(τ, x) ≥ min (0, min(h(0, x))) = −1, which implies that (3.22)
g(t, x) ≥ −e3(T −t)
for all (t, x) ∈ [0, T ] × R.
Exactly as in the proof of (3.22), we can also prove that (3.23)
g(t, x) ≤ e3(T −t)
for all (t, x) ∈ [0, T ] × R.
Combining (3.22) with (3.23), we get g(t, ·)L∞ ≤ e3(T −t) ,
(3.24)
0 ≤ t ≤ T.
With (3.24) and the similar proof to (3.3), we get (3.25)
|g(t, x)| ≤ exp[−1 (R + M (T − t) − |x|) + 4(T − t)ρ L∞ ]
for all (t, x) ∈ {(t, x) : |x| ≥ R + M (T − t), 0 ≤ t ≤ T }. On the other hand, taking ∂x to the first equation of (1.7), we have (3.26) ∂t (∂x ρ ) + ∂x (u S (ρ )∂x ρ ) + (S (ρ ) + ρ S (ρ ))∂x ρ − ∂xx ∂x ρ = 0. Combining (3.18), (3.25), and (3.26), we get ∂x ρ (T, x)γ(x) dx − ∂x ρ0, g(0, x) dx R
(3.27)
=
0
T
R
R
(∂t (∂x ρ )g + ∂x ρ ∂t g) dx dt = 0.
It follows from (3.24) that ∂x ρ (T, x)γ(x) dx ≤ e3T |∂x ρ0, | dx 2 R R R ≤ e3T (3.28) |dρ0 |, R
which implies (3.17). This completes the proof of the lemma. We now complete the proof of Theorem 1.2.
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QIANG DU AND PING ZHANG
Proof of Theorem 1.2. First by (3.17), we find that {ρ (t, x)} is uniformly bounded in L∞ ([0, T ], BV (R)). While by (3.1), (3.2), and the first equation in (1.7), we find that {∂t ρ } is uniformly bounded in L∞ ([0, T ], W −1,∞ (R)) + L2 ([0, T ], H −1 (R)). Notice by the compact embedding theorem that BV (R) >→>→ Lp (R) for any p < ∞. Thus by the Lions–Aubin lemma and a proof similar to that of Lemma 3 in [34], we find that there exists a ρ ∈ L∞ ([0, T ], BV (R)) for any T < ∞ such that (3.29)
ρ → ρ
in C([0, T ], Lp (K))
for all
T 0} and supp ρ0 =: {x, ρ0 (x) < 0}. Then m = |ρ| for + − almost all (t, x) ∈ D = D ∪ D , where − D+ = {(t, Φ+ t (x)) : x ∈ supp ρ0 } ,
+ D− = {(t, Φ− t (x)) : x ∈ supp ρ0 },
with Φ± t (x) being defined by (3.31)
dΦ± t (x) = ±u(t, Φ± t (x)), dt ± Φt (x)|t=0 = x.
Proof. First by multiplying signρ on both sides of (1.7), we find (3.32)
∂t |ρ | + div(u sign(ρ )(S (ρ ) − S (0))) ≤ −|ρ |S (0) + |ρ |.
Then by (3.1), it is trivial to prove that (3.33)
sign(ρ )(S (ρ ) − S (0)) − ρ L1 (K) → 0,
S (0)ρ L1 (K) → 0
as → 0. Hence by summing up (3.11), (3.32), (3.33), and taking → 0 in (3.32), we find ∂t m + div(uρ) ≤ 0.
(3.34)
Now let w = ρ + m. By summing up (3.15) and (3.34), we find ∂t w + div(uw) ≤ 0.
(3.35) Denote w (t, x) = satisfies (3.36)
j (y)w(t, x R2
− y) dy, and by [22, Lemma 2.3], we find that w
∂t w + div(uw ) ≤ R (t, x),
1295
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
where R (t, x) = div(uw ) − j ∗ div(uw), and R → 0 in L1loc (R+ × R2 ). Equation (3.36) directly implies that (3.37)
dw (t, Φ+ t (x)) + ≤ (ρw )(t, Φ+ t (x)) + R (t, Φt (x)). dt
On the other hand, by (3.3), we find that suppρ ⊂ {(t, x) : |x| ≤ r + M t} =: B. Thus by [21, Lemma 1] or [33], we find that (3.31) has a unique global solution such that (3.38)
C(T )−1 |x1 − x2 |e
4πt
+ e ≤ |Φ+ t (x1 ) − Φt (x2 )| ≤ C(T )|x1 − x2 |
−4πt
,
and because divu = ρ, by [10, equation (74)], we have (3.39)
ρL∞ t R L1 (BT ) → 0 R (t, Φ+ t (x))L1 (K) ≤ e
as → 0, where BT = B ∩ {(t, x) : t ≤ T }. By summing up (3.37), (3.39), and letting → 0 in (3.37), we get for almost all x ∈ R2 that there holds (3.40)
dw(t,Φ+ t (x)) ≤ (ρw)(t, Φ+ t (x)), dt + w(t, Φt (x))|t=0 = 0, x ∈ supp ρ− 0.
Then the Gronwall inequality implies that (3.41)
w(t, Φ+ t (x)) = 0,
x ∈ supp ρ− 0,
t ∈ R+ .
By (3.38) and (3.41), we get (3.42)
d(t, x) = −ρ(t, x)
for a.e. (t, x) ∈ D+ .
Similarly, by subtracting (3.15) from (3.34) and letting q = m − ρ, we have (3.43)
∂t q − div(uq) ≤ 0.
By the proof of (3.42), we have (3.44)
m(t, x) = ρ(t, x)
for a.e. (t, x) ∈ D− .
Combining (3.42) and (3.44), we complete the proof of the proposition. Appendix. The construction of the approximate solutions. Let S (ξ) = |ξ| ∗ j (ξ), where j (ξ) is the standard Friedrich mollifier with supp j (·) ⊂ B (0). In the following, we are going to prove the global existence of smooth solutions to the following equations: d ∂t ρ + div(uS (ρ)) = ρ, (t, x) ∈ (0, x∞) × R , d = 1, 2, 3, −1 u = ∇ ρ for d = 2, 3, u = −∞ ρ dy for d = 1, (A.1) ρ|t=0 = ρ0 .
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QIANG DU AND PING ZHANG
Theorem A.1. Let s > d/2 + 1, ρ0 ∈ H0s (Rd ). Equation (A.1) has a unique global solution (ρ, u) such that ρ, ∇u ∈ L∞ ([0, T ], H s (Rd )) ∩ L2 ([0, T ], H s+1 (Rd )) for any T < ∞. Furthermore, (A.2) (A.3)
ρ(t, ·)L1 ≤ ρ0 L1 , ρ(t, ·)L∞ ≤ ρ0 L∞ + . t ρ2 (t, x) dx + 2 |∇ρ|2 dx ds ≤ ρ20 dx + 2 t Rd
0
Rd
Rd
Rd
|ρ0 | dx.
Proof. Following the standard argument for a nonlinear parabolic equation, we can prove the local existence and uniqueness of solution (ρ, u) to (A.1) such that ρ, ∇u ∈ L∞ ([0, T ], H s (Rd )) ∩ L2 ([0, T ], H s+1 (Rd )) for some positive constant T. Now let T ∗ be the lifespan of the solution (ρ, u) to (A.1). Then for t < T ∗ , notice the classical convex inequality: signρρ ≤ |ρ|. By multiplying signρ on both sides of (A.1), we find ∂t |ρ| + div(u signρ(S (ρ) − S (0))) ≤ −|ρ|S (0) + |ρ|.
(A.4)
Integrating the above inequality over Rd , we get the first inequality of (A.2) for t < T ∗ . Next, multiplying pρp−1 on both sides of the first equation of (A.1) with p an even integer, we get ∂t ρp + div(uF (ρ)) = G (ρ) + pρp−1 ρ,
(A.5)
ρ with F (ρ) = 0 pS (ξ)ξ p−1 dξ and G (ρ) = ρF (ρ) − pρp S (ρ). By the definition of S (ρ), we have pρ 0 S (ξ)ξ p−1 dξ − (p − 1)ρp+1 − p ρ + pρp Rd ξj (ξ) dξ, ρ pρ 0 S (ξ)ξ p−1 dξ − pρp S (ρ), G (ρ) = − pρ 0 S (ξ)ξ p−1 dξ + (p − 1)ρp+1 + p ρ − pρp Rd ξj (ξ) dξ, which together with the simple inequalities that S (ξ)ξ p−1 dξ ≤ p , p|ρ|p ≤ (p − 1)|ρ|p+1 + |ρ|, p
ρ ≥ , |ρ| ≤ ρ ≤ −,
S (ρ) ≥ 0
0
gives us G (ρ) ≤ p |ρ|.
(A.6)
Combining (A.2) with (A.6), and integrating (A.5) over Rd , we find t ρp dx + p(p − 1) ρp−2 |∇ρ|2 dx ds d d 0 R R t p p ρ0 dx + |ρ| dx ds ≤ ρp0 dx + p t ≤
(A.7)
Rd
0
Rd
Rd
Rd
|ρ0 | dx.
In particular, by taking p = 2 in the above, we get (A.3). Moreover, for any compact subset K of Rd , (A.7) implies that (A.8)
K
p
ρ (t, x) dx
p1
1
1
≤ ρ(t, ·)Lp ≤ ρ0 Lp + t p ρ0 Lp 1 .
EXISTENCE OF SOLUTIONS FOR VORTEX DENSITY MODELS
1297
Letting p → ∞ in (A.8), we prove the second inequality of (A.2) for t < T ∗ . Now let E(t, x) be the fundamental solution of the heat operator (∂t − ); then d
|x|2
E(t, x) = (4πt)− 2 e− 4t ,
(A.9)
t > 0,
by (A.1), and ρ can also be written by the following form: ρ(t, x) = −
t Rd
0
E(t − s, x − y)div(uS (ρ))(s, y) dy ds +
Rd
E(t, x − y)ρ0 (y) dy
for t < T ∗ . By integrating by parts in the above formula, we find t (A.10) ρ(t, x) = − ∇x E(t − s, x − y)(uS (ρ))(s, y) dy ds + ρ0 (t, x), Rd
0
and consequently, ∂x ρ(t, x) = −
t Rd
0
(A.11)
∇x E(t − s, x − y)(∂x uS (ρ) + uS (ρ)∂x ρ)(s, y) dy ds + ∂x ρ0 (t, x).
Thus by the Hausdorff–Young inequality and the fact that |S (ξ)| ≤ 1, we get ∂x ρ(t, ·)L1 ≤ C(sup ρ(t, ·)L1 + uL∞ ) t>0
(A.12) ∂x ρ(t, ·)L∞
0
t
1
(t − s)− 2 (∂x u(s, ·)L∞
+ ∂x ρ(s, ·)L1 ) ds + ∂x ρ0 L1 , t 1 ≤ C(ρ(t, ·)L∞ + uL∞ ) (t − s)− 2 (∂x u(s, ·)L∞ 0
(A.13)
+ ∂x ρ(s, ·)L∞ ) ds + ∂x ρ0 L∞ .
On the other hand, for d = 2 and 3, by the second equation of (A.1), we have x−y |u(t, x)| = ρ(t, y) dy d Rd |x − y| 1 ρ(t, ·)L1 ≤ ρ(t, ·)L∞ dy + d−1 Rd−1 |x−y|≤R |x − y| (A.14)
≤ CRρ(t, ·)L∞ + ρ(t,·)
ρ(t, ·)L1 . Rd−1
1
1 Taking R = ( ρ(t,·)LL∞ ) d in (A.14), we find 1
1− 1
(A.15) u(t, ·)L∞ ≤ Cρ(t, ·)Ld 1 ρ(t, ·)L∞d ≤ C (ρ(t, ·)L1 + ρ(t, ·)L∞ ) , while for d = 1, the second equation of (A.1) directly implies that (A.16)
u(t, ·)L∞ ≤ ρ(t, ·)L1 .
Similar to the proof of (A.15) and (A.16), we have (A.17)
∂x u(t, ·)L∞ ≤ C (∂x ρ(t, ·)L1 + ∂x ρ(t, ·)L∞ ) .
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QIANG DU AND PING ZHANG
Now let us set y(t) = ∂x ρ(t, ·)L1 + ∂x ρ(t, ·)L∞ . By combining (A.2) with (A.12)– (A.17), we find y(t) ≤ C
0
t
1
(t − s)− 2 y(s) ds + y0 ,
t < T ∗.
The Gronwall inequality yields that y(t) ≤ y0 eC
(A.18)
√
t
,
t < T ∗.
On the other hand, standard energy estimates (see [24]) show that if T ∗ < ∞, then (A.19)
lim (∂x u(t, ·)L∞ + ∂x ρ(t, ·)L∞ ) = ∞.
t→T ∗
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