ON THE REGULARITY OF SOLUTIONS TO A PARABOLIC SYSTEM ...
ON THE REGULARITY OF SOLUTIONS TO A PARABOLIC SYSTEM RELATED TO MAXWELL’S EQUATIONS KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
Abstract. The goal of this paper is to establish H¨ older estimates for the solutions of a certain parabolic system related to Maxwell’s equations. Such an estimate is employed to get the local H¨ older continuity of the magnetic field arising from Maxwell’s equations in a quasi-stationary electromagnetic field, provided the resistivity of the material is continuous in time.
1. Introduction Let Ω be a domain in R3 and let Q = Ω × (0, T ) be a cylinder in R4 . Let A(x, t) be a 3 × 3 symmetric matrix such that there exists a number ν ∈ (0, 1) satisfying (1.1)
2
ν |ξ| ≤ hA(x, t)ξ, ξi
and
|A(x, t)| ≤ ν −1
∀(x, t) ∈ Q.
Here hξ, ηi denotes the usual inner product of vectors ξ, η ∈ R3 and |X| is the 2 Euclidean length of X ∈ RN , i.e., |X| is the sum of squares of each entry of X. In this paper we study the regularity of the solutions of the following system: ut + ∇×[A(x, t)∇×u] = 0 (1.2) in Q. ∇·u=0 Here, we denote ∇×u = curl u and ∇ · u = div u, where u = (u1 , u2 , u3 ) ∈ R3 . Recently, in [3], the elliptic case of this system has been studied by the first two authors. Without imposing any assumptions other than (1.1), they derived a priori H¨ older estimate of its weak solutions. Although a special type of A(x) was considered in the article [3], the main result [3, Theorem 2.1] remains valid as long as A(x) satisfies (1.1). We would like to mention that independently, Yin obtained a similar result in [12]. In this context, it is an interesting question to ask whether or not the weak solutions of the system (1.2) are locally H¨older continuous when the coefficients A(x, t) are assumed to be only measurable. Our main result states that weak solutions of the system (1.2) are locally H¨older continuous in the case that A(x, t) is uniformly continuous in time or A(x, t) is of the form a(t)B(x) where a(t) is a scalar function and B(x) is a symmetric matrix (see Theorem 3.2 and Theorem 3.3 below). The main idea is to use the elliptic result of [3], as well as standard perturbation techniques and the structure of the system (1.2). As an application of the main result, we consider the following system arising from Maxwell’s equations in a quasi-stationary electro-magnetic field where the The first author was supported in part by NSF Grant No. DMS-9877055. The second author was supported in part by NSF Grant No. DMS-9971052. 1
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
displacement of the electric current is assumed to be time independent: H t + ∇×[%(x, t)∇×H] = 0 in Q. ∇·H =0 Here, the vector H represents the strength of the magnetic field and % the electrical resistivity of the material (see [5], [10]). Using our result from Section 3, one finds that if % is assumed to be continuous in time, then H is locally H¨older continuous. Another application is the following coupled system introduced in [9]: H t + ∇×[%(u)∇×H] = 0 ∇·H =0 (1.3) in Q = Ω × (0, T ), 2 ut − ∆u = %(u) |∇×H| where H and u are the magnetic field and the temperature of the material, respectively, and %(u) is the electrical resistivity of the material. In [10] Yin proved the H¨ older continuity of (H, u) under the assumption that %(u) is continuous. Using our result of Section 3, the same conclusion holds, only assuming the continuity of %(u) in time. Finally we consider the Stokes system with measurable coefficients: ut − A(x, t)∆u + ∇p = f in Q = R3 × (0, ∞). div u = 0 With the same assumption on A(x, t) as in Section 3, we prove that u is locally H¨ older continuous. This paper is organized as follows: In Section 2, we introduce the notations and recall some known results used in our proofs. In Section 3, we study the system (1.2) and prove our main result. In Section 4, we study some parabolic systems related to Maxwell’s equations and the Stokes system with measurable coefficients, as applications. 2. Notations and Preliminaries In this section, we introduce the notations which will be used throughout this article and also recall some well-known facts. Let us begin with the notations. • z0 = (x0 , t0 ) denotes an arbitrary point in Rn+1 , where x0 ∈ Rn and t0 ∈ (−∞, ∞). • Br = Br (x0 ) = {x ∈ Rn : |x − x0 | < r}. • Qr = Qr (z0 ) = (x, t) ∈ Rn+1 : |x − x0 | < r, −r2 < t − t0 < 0 . • Qr,t = Qr,t (z0 ) = {(x, t) ∈ Qr (z0 )}; i.e., Qr,t (z0 ) = Br (x0 ) × {t} if t ∈ (t0 − r2 , t0 ) and Qr,t (z0 ) = ∅ otherwise. • For Q0 ⊂ Q, ∂p Q0 is the parabolic boundary of Q0 . • Q0 b Q means RQ0 is compact and Q0 ∪ ∂p Q0 ⊂ Q.R R R • We denote by S f Rthe average of f on S; i.e., S f = S f / S 1 and we denote fr = fz0 ,r = Qr (z0 ) f . • For Ω ⊂ Rn and q > 1, Lq (Ω) denotes the Banach space of measurable functions with the following norms: Z 1/q q kukLq (Ω) = |u(x)| dx and kukL∞ (Ω) = ess sup |u| . Ω
Ω
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
3
• Let Q = Ω × (a, b). For 1 ≤ q, r ≤ ∞, Lq,r (Q) denotes the Banach space of all measurable functions with the finite norm r/q !1/r Z b Z q kukq,r; Q = kukLq,r (Q) = |u(x, t)| dx dt . a
q,q
Ω
q
• L (Q) will be denoted by L (Q) and the norm k · kLq,q (Q) by k · kLq (Q) . • For 1 ≤ q ≤ ∞, W k,q (Ω) denotes the usual Sobolev space; i.e., W k,q (Ω) = {u : Dα u ∈ Lq (Ω), 0 ≤ |α| ≤ k}, and W0k,q (Ω) denotes the completion of C0∞ (Ω) in W k,q (Ω). • For 1 ≤ q, r ≤ ∞, Lr (a, b); W k,q (Ω) denotes the Banach space of all measurable functions with the finite norm !1/r Z b r kukLr ((a,b);W k,q (Ω)) = ku(·, t)kW k,q (Ω) dt . a
• C α,α/2 (Q) denotes the Banach space of functions that are H¨older continuous with the exponent α ∈ (0, 1), and |u(z) − u(z 0 )| , d(z, z 0 )α z6=z 0 ∈Q
[u]α,α/2; Q = [u]C α,α/2 (Q) = sup
1/2
where d(·, ·) is the parabolic metric; i.e., d(z, z 0 ) = |x − x0 | + |t − t0 | . α,α/2 • u ∈ Lp,q ) means u ∈ Lp,q (Q0 ) (resp., u ∈ C α,α/2 (Q0 )) for loc (resp., u ∈ Cloc 0 0 all Q = Ω × (a, b) b Q. • The Morrey space M 2,µ (Q) is defined to be the set of all functions u ∈ L2loc (Q) with the finite norm !1/2 Z kukM 2,µ (Q) =
sup
2
ρ−µ
Qρ (z)⊂Q
|u|
.
Qρ (z)
• We denote by N = N (α, β, . . .) a constant depending on the prescribed quantities α, β, . . .. The following lemma is Campanato’s integral characterization of H¨older continuous functions (see e.g. [6, Lemma 4.3, page 50]). Lemma 2.1. Let fR ∈ L2 (Q2R (z0 )) and suppose there are positive constants α ≤ 1 2 and H such that Qr (z) |f − fz,r | ≤ H 2 rn+2+2α for any z ∈ QR (z0 ) and any r ∈ (0, R). Then f is H¨ older continuous with the exponent α in QR (z0 ) and [f ]α,α/2;QR (z0 ) ≤ N (n, α)H. The following lemma can be found in Giaquinta [1, Lemma 2.1, page 86]. Lemma 2.2. Let φ(t) be a nonnegative and nondecreasing function. Suppose that i h ρ α + ε φ(r) + Brβ φ(ρ) ≤ A r for all ρ < r ≤ R, with A, α, β nonnegative constants, β < α. Then there exists a constant ε0 = ε0 (A, α, β) such that if ε < ε0 , for all ρ < r ≤ R we have ρ β β φ(ρ) ≤ c φ(r) + Bρ r where c is a constant depending on α, β, A.
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
3. Main result This section deals with the linear theory and a priori estimates. To avoid the technicalities, the coefficient A(x, t) will be assumed to be smooth as in [3]. Nonetheless, the constant appearing in the a priori estimates will not depend on the extra smoothness of A(x, t). Also, by a solution, we always mean a smooth solution unless otherwise stated. Indeed, the energy inequality below indicates that our system (1.2) is strongly parabolic. Hence, the general theory on the parabolic systems of divergence type can be applied, and by requiring A(x, t) to be smooth, we easily see that the weak solutions of (1.2) are actually smooth (see, e.g. [4]). Lemma 3.1 (Energy inequality). Let u be a solution of (1.2). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then the following estimate holds: Z Z Z N (ν, λ) 2 2 2 sup |u(·, s)| + |∇u| ≤ |u| . 2 2 R t0 −R ≤s≤t0 BR QR QλR Proof. Assume, for simplicity, that z0 = (0, 0). Let η be a cut-off function which vanishes near ∂p QλR . Using η 2 u as a test function, we find (see e.g. [4] or [6]) Z Z Z 2 2 2 2 2 2 sup η |u(·, s)| + η |∇×u| ≤ N (ν) |ηt | + |∇η| |u| . −R2 ≤s≤0
Ω
Q
Q
On the other hand, from the vector identity ∇×∇×u = −∆u + ∇(∇ · u)
(3.1)
and the equation ∇ · u = 0, we find Z Z Z 2 2 2 2 2 2 η |∇u| ≤ N η |∇×u| + |∇η| |u| . Q
Q
Q
Combining the above inequalities and choosing a suitable η, we finish the proof. Next lemma states that if A(x, t) does not depend on time; i.e., A = A(x), then any solution u of (1.2) satisfies ut ∈ L2loc (Q) and ∇u ∈ L2,∞ loc (Q). Lemma 3.2. Let u be a solution of (1.2) with A = A(x). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then, we have Z Z Z N (ν, λ) 2 2 2 |ut | + sup |∇u(·, s)| ≤ |u| . R4 t0 −R2 ≤s≤t0 BR QR QλR Proof. As before, we assume z0 = (0, 0). Let η be a cut-off function vanishing near ∂p Q√λR . Using η 2 ut as a test function, we find (see, e.g. [4]) Z Z Z 2 2 2 2 η 2 |ut | + sup η 2 |∇×u(·, s)| ≤ N (ν) (|ηt | + |∇η| ) |∇u| . −R2 ≤s≤0
Q
Ω
Q
Also, from (3.1) we find that Z Z Z 2 2 2 2 η 2 |∇u(·, s)| ≤ N η 2 |∇×u(·, s)| + |∇η| |u(·, s)| . Ω
Ω
Ω
Combining the√above inequalities and applying Lemma 3.1 with QR and λ replaced by Q√λR and λ, respectively, we complete the proof. Next lemma says that if A = A(x), then ut belongs to L2,∞ loc (Q). This is where the result of the elliptic case proved in [3] is applied.
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5
Lemma 3.3. Let u be a solution of (1.2) with A = A(x). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then, the following estimate holds: Z Z Z N (ν, λ) 2 2 2 sup |ut (·, s)| + |∇ut | ≤ |u| . 6 2 R t0 −R ≤s≤t0 BR QR Q Proof. As before, we assume z0 = (0, 0). Taking the derivatives with respect to time in (1.2), we get utt + ∇×(A∇×ut ) = 0,
∇ · ut = 0.
Applying Lemma 3.1 to ut , and then using Lemma 3.2 we find Z Z Z Z N (ν, λ) N (ν, λ) 2 2 2 2 sup |ut (·, s)| + |∇ut | ≤ |ut | ≤ |u| . 2 6 2 R R −R ≤s≤0 BR QR Q√λR QλR This completes the proof.
The following lemma is adapted from Struwe [8]. One can easily verify it by following the proof of Lemma 3 in [8] line by line. Lemma 3.4. Let u be a solution of (1.2). Let R > 0 be such that QλR = QλR (z0 ) ⊂ Q for some λ > 1. Then, there exist a constant N = N (ν, λ) such that Z Z 2 2 (3.2) |u − uR | ≤ N R2 |∇u| . QR
QλR
The next theorem is our first main result. The idea is to treat ut as an inhomogeneous term and apply the elliptic result of [3]. This kind of approach is found, for example, in [7]. Theorem 3.1. Let u be a solution of (1.2) with A = A(x). Then, u is locally H¨ older continuous in Q. Moreover, the following estimate holds: for any R > 0 such that Q6R = Q6R (z0 ) ⊂ Q, [u]C α,α/2 (QR ) ≤ N (ν)R−(5+2α) kukL2 (Q6R ) , where α = α(ν) ∈ (0, 1/2]. Proof. We assume that R = 1 and z0 = (0, 0). The general case is recovered by a simple coordinate change (x, t) 7→ ((x − x0 )/R, (t − t0 )/R2 ). From Lemma 3.3, we know that ut ∈ L2,∞ (Q5 ). Also, we find that ∇ · ut = 0 from the vector identity ∇ · ∇×F = 0. Hence, we can rewrite (1.2) as ∇×(A(x)∇×u) = −ut
(3.3)
and apply the elliptic result [3, Theorem 2.1] to find that for any t ∈ (−42 , 0), we have (3.4) [u(·, t)]C α (Q4,t ) ≤ N (ν) ku(·, t)kL2 (Q5,t ) + kut (·, t)kL2 (Q5,t ) , where α = α(ν) ∈ (0, 1/2] (see Remark 3.1 below). From Lemma 3.1 and Lemma 3.3, the right hand side of (3.4) is uniformly bounded for t ∈ (−42 , 0) hence (3.5)
[u(·, t)]C α (Q4,t ) ≤ N (ν) kukL2 (Q6 )
∀t ∈ (−42 , 0).
Fix z = (x, t) ∈ Q1 and r ≤ 1. From Lemma 3.4, we have Z Z 2 2 (3.6) |u − uz,r | ≤ N r2 |∇u| . Qr (z)
Q2r (z)
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
From (3.3), the following Caccioppoli type inequality holds (see [3, Lemma 4.4]). ! Z Z 1 2 2 2 |u − u(t)| + kut (·, t)kL6/5 (Q3r,t (z)) , |∇u| ≤ N r2 Q3r,t (z) Q2r,t (z) R where u(t) = Q3r,t (z) u(·, t). Note that, from (3.5), Z 1 2 2 |u − u(t)| ≤ N r1+2α [u(·, t)]2C α (Q4,t ) ≤ N r1+2α kukL2 (Q6 ) . r2 Q3r,t (z) Also, from H¨ older’s inequality and Lemma 3.3 2
2
2
kut (·, t)kL6/5 (Q3r,t (z)) ≤ N r2 kut (·, t)kL2 (Q3r,t (z)) ≤ N r2 kukL2 (Q6 ) . Therefore, we have Z 2 |∇u| Q2r (z)
=
Z
t
t−4r 2
Z
2
|∇u(y, s)| dyds
Q2r,t (z)
2 2 ≤ N r2 r1+2α + r2 kukL2 (Q6 ) ≤ N r3+2α kukL2 (Q6 ) , where we used α ≤ 1/2 in the last step. Hence, using (3.6) we conclude Z 2 2 |u − uz,r | ≤ N (ν)r2α kukL2 (Q6 ) , ∀r ≤ 1, ∀z ∈ Q1 . Qr (z)
Finally, from Lemma 2.1, we get [u]α,α/2,Q1 ≤ N (ν) kukL2 (Q6 ) . This completes the proof. Remark 3.1. In [3, Theorem 2.1], A(x) is assumed to be of the form a(x)I. However, the proof works for general A(x) as long as A(x) satisfies (1.1). Also, one can find in the proof that α ≤ γ, where γ = 2 − 3/2 = 1/2 in our case. Definition 3.1. Let f : Q → RN be measurable. For all r > 0 and z = (x, t) such that Qr (z) ⊂ Q, we define ωf (r; z) := ess sup(y,s)∈Qr (z) |f (y, t) − f (y, s)| and (3.7)
ωf (r) := sup {ωf (r; z) : ∀z ∈ Q such that Qr (z) ⊂ Q} .
Lemma 3.5. Let u be a solution of (1.2). Suppose there is a fixed τ ∈ (0, 1) such that for all R > 0 satisfying QR ⊂ Q, [u]C α,α/2 (Qτ R ) ≤ N R−(5+2α)/2 kukL2 (QR ) .
(3.8)
Then for all 0 < ρ ≤ r ≤ R, we have the following estimate for ∇u: Z ρ 3+2α Z 2 2 |∇u| ≤ N (τ ) (3.9) |∇u| . r Qr Qρ Proof. We may assume ρ < (τ /4)r, for (3.9) is obvious if ρ ≥ (τ /4)r. We denote [u]α,r := [u]C α,α/2 (Qr ) . Applying the energy inequality to u − u2ρ , which is also a solution of (1.2), we find (recall 2ρ ≤ (τ /2)r) Z Z 2 2 −2 (3.10) |∇u| ≤ N ρ |u − u2ρ | ≤ N ρ3+2α [u]2α,(τ /2)r . Qρ
Q2ρ
From (3.8) applied to v = u − ur/2 , and by Lemma 3.4, Z Z u − ur/2 2 ≤ N r−3−2α [u]2α,(τ /2)r = [v]2α,(τ /2)r ≤ N r−5−2α Qr/2
Putting the above inequalities together, we complete the proof.
2
|∇u| .
Qr
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Theorem 3.2. Let u be a solution of (1.2) with A(x, t) satisfying: (A) limρ→0 ωA (ρ) = 0, where ωA is defined as in Definition 3.1. Then, u is locally H¨ older continuous in Q. More precisely, let β ∈ (0, α), where α is the H¨ older exponent in Theorem 3.1, and let Q0 b Q. Then, there is Q00 such that Q0 ⊂ Q00 b Q and [u]C β,β/2 (Q0 ) ≤ N (ν, β, ωA , Q0 , Q) kukL2 (Q00 ) . Proof. Let R0 be a fixed number which will be specified later. Fix z0 ∈ Q0 and let R > 0 be such that 2R ≤ R0 and Q4R = Q4R (z0 ) ⊂ Q. We will show that [u]C β,β/2 (QR ) ≤ N (ν, β, R) kukL2 (Q4R ) . Then, the theorem will follow from a standard covering argument. Fix z = (x, t) ∈ QR and let 0 < ρ < r ≤ R. For any z 0 = (y, t) ∈ QR , denote A0 (y) = A(y, t) and let v be a solution of the system v t + ∇×[A0 ∇×v] = 0,
∇ · v = 0 in Q2r (z)
with the boundary condition v = u on ∂p Q2r (z). Let α = α(ν) be the H¨ older exponentR from Theorem 3.1. From (3.9) applied to R 2 2 v, we have Q2ρ (z) |∇v| ≤ N (ρ/r)3+2α Q2r (z) |∇v| , and hence, # " Z Z ρ 3+2α Z 2 2 2 |∇u| + |∇(u − v)| . |∇u| ≤ N r Q2r (z) Q2r (z) Q2ρ (z) Using the equations satisfied by u and v, the function w = u − v satisfies wt + ∇×[A0 ∇×w] = ∇×[(A0 − A)∇×u],
∇ · w = 0 in Q2r (z)
and w = 0 on ∂p Q2r (z). By using w itself as a test function to the above equation R R 2 2 we find Q2r (z) |∇w| ≤ N ωA (2r) Q2r (z) |∇u| (recall ∇ · w = 0) and hence # " Z Z ρ 3+2α Z 2 2 2 |∇u| ≤ N |∇u| + ωA (2r) |∇u| . r Q2ρ (z) Q2r (z) Q2r (z) Now, choose R0 such that ωA (R0 ) < ε0 , where ε0 is as in Lemma 2.2. From Lemma 3.4, Lemma 2.2, and Lemma 3.1, we get Z Z 2 2 |u − uz,r | ≤ N (ν)r2 |∇u| Qr (z) Q2r (z) Z Z 2 2 ≤ N (ν, β, R)r5+2β |∇u| ≤ N (ν, β, R)r5+2β |u| . Q2R
Therefore, the theorem follows from Lemma 2.1.
Q4R
Let A(x, t) ≡ a(t)B(x), where a(t) is a positive, bounded scalar function and B is a symmetric matrix satisfying (1.1). The hypothesis (A) in Theorem 3.2 is not satisfied in this case unless a(t) is assumed to be continuous. However, in such a case, it turns out that we don’t need to impose any continuity assumptions on a(t) to get the H¨ older estimate for u, as we will show next. Theorem 3.3. Let u be a solution of (1.2) with A(x, t) = a(t)B(x), where a(t) is a scalar function such that ν ≤ a(t) ≤ ν −1 and B(x) is a symmetric matrix satisfying (1.1). Then, for any R > 0 such that Q7R = Q7R (x0 ) ⊂ Q, u is H¨ older continuous in QR and (3.11)
[u]C α,α/2 (QR ) ≤ N (ν)R−(5+2α)/2 kukL2 (Q7R ) ,
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
where α = α(ν) ∈ (0, 1/2]. Proof. We may assume z0 = (0, 0) and R = 1 as before. From the assumption that A = a(t)B(x), we see that (1.2) becomes (3.12)
ut + a(t)∇×[B(x)∇×u] = 0,
∇ · u = 0.
2 −1
Using η a But as a test function in (3.12) and proceeding as in the proof of Lemma 3.2, we obtain the estimate Z Z 2 2 2 (3.13) |ut | + sup |∇u(·, s)| ≤ N (ν) kukL2 (Q7 ) . −62 ≤s≤0
Q6
B6
2
In particular, we have ut ∈ L (Q6 ). Next, by taking the curl in the equation (1.2) and denoting w = ∇×u, we get (3.14)
wt + a∇×[∇×Bw] = 0. 2 −1
Using η a (3.15)
Z
Bwt as a test function in (3.14) and proceeding as above, we find Z Z 2 2 2 |wt | + sup |∇×Bw| ≤ N (ν) |∇×Bw| . −52 ≤s≤0
Q5
B5
Q6
−1
RSince ∇×Bw2 = −a ut from (3.12), the right hand side of (3.15) is estimated as |∇×Bw| ≤ N (ν) kukL2 (Q7 ) from (3.13). Hence, it follows that Q6 2
kut kL2,∞ (Q5 ) ≤ N (ν) k∇×BwkL2,∞ (Q5 ) ≤ N (ν) kukL2 (Q7 ) . The rest of proof goes exactly as in that of Theorem 3.1, and therefore, the details are omitted. Corollary 3.1. Let β ∈ (0, α), where α is the H¨ older exponent in Theorem 3.3. Suppose u is a solution of (1.2) with A(x, t) satisfying the following assumption: there exists A0 (x, t) = a(t)B(x) such that (A0 ) |A(z) − A0 (z)| ≤ ε for all z ∈ Q2R ⊂ Q, where a(t) and B(x) satisfy the assumptions in Theorem 3.3. Then, there exists ε0 = ε0 (ν, β) such that if ε < ε0 , u is H¨ older continuous in QR and [u]C β,β/2 (QR ) ≤ N (ν, β, R) kukL2 (Q2R ) . Proof. From Theorem 3.3 and Lemma 3.5, we see that the estimate (3.9) holds. Hence, the proof of Theorem 3.2 works here. Remark 3.2. We may also consider the system (1.2) with an inhomogeneous term: ut + ∇×[A(x, t)∇×u] = ∇×f (3.16) in Q. ∇·u=0 Let β ∈ (0, α), where α is the H¨older exponent in Theorem 3.1. Suppose f ∈ M 2,3+2β (Q) and A(x, t) satisfies (A) or (A0 ). Let u be a solution of (3.16). It can be easily verified, by modifying the proof of Theorem 3.2, that u is H¨older continuous in Q0 b Q and that it satisfies the following estimate: [u]C β,β/2 (Q0 ) ≤ N (ν, β, Q0 , Q) kukL2 (Q) + kf kM 2,3+2β (Q) .
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4. Applications We first consider a certain system arising in a Maxwell’s equations. Let a conductive material occupy a bounded domain Ω ⊂ R3 . Let E and H be the electric and the magnetic fields in Ω. Under certain assumptions, Yin derived the following mathematical model (see [9], [11]): (E)t + σE = ∇×H, (µH)t + ∇×E = 0, (4.1) in Q, div(µH) = 0, where , µ, and σ are the electric permittivity, the magnetic permeability and the electric conductivity of the material, respectively. If the electrical displacement D = E is negligible and µ is a constant, the above system can be reduced to H t + ∇×[%(x, t)∇×H] = 0, (4.2) in Q, div H = 0, where % is the resistivity of the material (see [9], [10]). The above system is the special case of the system (1.2), where A = %(x, t)I and I is the identity matrix. Therefore, our result implies that H is locally H¨older continuous in Q provided that %(x, t)I satisfies the assumption (A) or (A0 ) of the previous section. Theorem 4.1. Let H be a weak solution of (4.2). Suppose that %I satisfies either (A) or (A0 ). Then H is locally H¨ older continuous. In fact, the following estimate holds: for R > 0 such that Q2R = Q2R (z0 ) ⊂ Q, there exists α ∈ (0, 1/2) such that [H]C α,α/2 (QR ) ≤ N kHkL2 (Q) . Next, we consider the case when temperature affects the electrical resistance. By taking the temperature effect into consideration, Yin derived the following mathematical model (see Yin [9], [10]): H t + ∇×[%(u)∇×H] = 0 ∇·H =0 (4.3) in Q = Ω × (0, T ), 2 ut − ∆u = %(u) |∇×H| where H and u are the magnetic field and the temperature of the material, respectively, and %(u) is the electrical resistivity of the material. Let us assume that (4.4)
lim ω%(u) (r) = 0,
r→0
where ω%(u) (r) is defined as in Definition 3.1. The assumption (4.4) above is an analogue of the assumption (A) in the previous section. Roughly speaking, this assumption is to say that the resistivity %(u) varies continuously in time, uniformly throughout the material Ω. For instance, the assumption (4.4) is satisfied if the scalar function % is Lipschitz continuous and the temperature u is continuous in time variable so that limr→0 ωu (r) = 0. We will show that, under the assumption (4.4), a pair of weak solution (H, u) of the system (4.3) is locally H¨older continuous. We remark that in [10], the continuity of %(u) in both spatial and time directions is assumed to get the same result. Theorem 4.2. Let (H, u) be a pair of weak solutions of (4.3). Suppose that %(u) satisfies (4.4). Then (H, u) is locally H¨ older continuous.
10
KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
The following lemmas will be used in the proof of Theorem 4.2 above. The proof of Lemma 4.1 below is standard (see, e.g. [8, Lemma 3]) and it will be omitted. Lemma 4.1. Let u be a solution of ut − ∆u = ∇ · f + g
(4.5)
in
Q.
Then the following estimate holds: for R > 0 such that Q2R ⊂ Q Z Z Z Z 2 2 2 2 |u − uR | ≤ N R2 |∇u| + R2 |f | + R4 |g| . QR
Q2R
Q2R
Q2R
Lemma 4.2. Let u be a smooth solution of (4.5) in Q and let R > 0 such that Q2R ⊂ Q. Assume that f ∈ M 2,3+2α (Q2R ) and g ∈ M 2,1+2α (Q2R ). Then, ∇u ∈ M 2,3+2α (QR ) and 2 2 2 2 k∇ukM 2,3+2α (QR ) ≤ N (R) kukL2 (Q2R ) + kf kM 2,3+2α (Q2R ) + kgkM 2,1+2α (Q2R ) . Sketch of proof. Fix z ∈ QR and r ∈ (0, R). Let v be the solution of vt − ∆v = 0 in Qr (z), v = u on ∂p Qr (z). R 2 2 Then, Qρ (z) |∇v| ≤ N (ρ/r)5 Qr (z) |∇v| for all ρ ≤ r. Let w := u − v. Note that w satisfies wt − ∆w = ∇ · f + g in Qr (z) and w = 0 on ∂p Qr (z). 2 2 Denote F = kf kM 2,3+2α (Q2R ) and G = kgkM 2,1+2α (Q2R ) . One can easily check R 2 that Qr (z) |∇w| ≤ N (F + G)r3+2α . Then, using Lemma 2.2, one may conclude R R 2 2 that Qr (z) |∇u| ≤ N [(r/R)3+2α QR |∇u| + (F + G)r3+2α ]. On the other hand, R R R R 2 2 2 2 from the energy inequality, QR |∇u| ≤ N (R−2 Q |u| + Q2R |f | + R2 Q2R |g| ). R 2 2 Therefore, we derived r−3−2α Qr (z) |∇u| ≤ N (R){kuk2;Q2R + F + G}. R
Now we are ready to prove Theorem 4.2. Proof of Theorem 4.2. We may assume kHk2 = kHk2;Q < ∞. Fix z0 ∈ Q and let R > 0 be such that Q8R = Q8R (z0 ) b Q. It is enough to show that (H, u) is H¨ older continuous in QR . From Theorem 3.2, there is 0 < α < 1/2 such that [H]C α,α/2 (Q8R ) ≤ N kHk2 , where N is a constant depending on the given data. Thus, using the triangle inequality, we find that H is locally bounded and kHkL∞ (Q4R ) ≤ N kHk2 .
(4.6)
It remains to show that u is H¨older continuous in QR . Using the following vector identity ∇ · (F × G) = (∇×F ) · G − F · (∇×G) and (4.3), we observe that 2
%(u) |∇×H| = ∇ · [H × (%(u)∇×H)] + H · H t , and thus the last equation of (4.3) can be rewritten as (4.7)
ut − ∆u = ∇ · [H × (%(u)∇×H)] + H · H t =: ∇ · f + g.
Fix z ∈ QR and r ∈ (0, R). Using (4.6) and proceeding as in (3.10) in the proof of Lemma 3.5 Z Z 2 2 2 4 |f | ≤ N kHk2 (4.8) |∇H| ≤ N r3+2α kHk2 . Q2r (z)
Q2r (z)
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
11
Similarly, using (4.6) and applying Lemma 3.3 to H (recall 2α ≤ 1), Z Z t Z 2 2 2 4 (4.9) |g| ≤ N kHk2 |H t | ≤ N r1+2α kHk2 . t−4r 2
Q2r (z)
Q2r (z)
Now, we apply Lemma 4.1 to (4.7) in Q2r (z) using (4.8) and (4.9) Z Z 2 2 4 |u − uz,r | ≤ N r2 |∇u| + N r5+2α kHk2 . Qr (z)
Q2r (z)
Hence, taking the average Z Z 2 (4.10) |u − uz,r | ≤ N r−3 Qr (z)
Q2r (z)
2
4
|∇u| + N r2α kHk2 .
Applying Lemma 4.2 to (4.7) and using (4.8) and (4.9), we conclude Z 2 2 4 −3−2α (4.11) r |∇u| ≤ N kukL2 (Q4R ) + kHk2 . Q2r (z)
Finally, by combining (4.10) and (4.11), we have Z 2 2 2 |u − uz,r | ≤ N r2α kukL2 (Q4R ) + kHkL2 (Q) . Qr (z)
Therefore, [u]α,α/2;QR < ∞. This completes the proof.
Next we consider the time-dependent Stokes system with measurable coefficients in diffusive term ut − A(x, t)∆u + ∇p = f (4.12) in Q ≡ R3 × (0, ∞). div u = 0 Here u and p are the velocity field and the pressure associated with u and f is a given external force. A(x, t) is a 3 × 3 symmetric matrix satisfying the ellipticity condition (1.1). The following theorem is the last application of our results from Section 3. Theorem 4.3. Let f ∈ M 2,3+2β (Q), where 0 < β < α and α is the H¨ older exponent in Theorem 3.1. Suppose ut , ∇p ∈ L2 (Q) and u ∈ L2 (0, ∞); W 2,2 (R3 ) such that u solves the system (4.12). Suppose that A(x, t) satisfies (A) or (A0 ). Then ∇u is locally H¨ older continuous and satisfies the following estimate: for R > 0 such that QR = QR (z0 ) ⊂ Q, we have [∇u]C β,β/2 (QR ) ≤ N k∇ukL2 (Q) + kf kM 2,3+2β (Q) , where N = N (ν, R). Proof. By taking the curl in (4.12), we have the following equations: wt + ∇ × (A(x, t)∇×w) = ∇ × f ,
w = ∇×u
in Q,
where we used the identity ∆u = −∇×∇×u + ∇(∇ · u). Then it is easy to see that w is locally H¨ older continuous in Q and the following estimate holds: [w]C β,β/2 (Q2R ) ≤ N (kwkL2 (Q4R ) + kf kM 2,3+2β (Q4R ) ), where N = N (ν, R). Using the Biot-Savart law, u can be recovered in terms of w as follows: Z (x − y) × w(y, t) 1 dy. u(x, t) = − 3 4π R3 |x − y|
12
KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT β,β/2
It is easy to see that ∇u ∈ Cloc [∇u]C β,β/2 (QR )
(Q) and the following estimate holds: ≤ N [w]C β,β/2 (Q2R ) ≤ N k∇ukL2 (Q) + kf kM 2,3+2β (Q) .
This completes the proof.
References
[1] Giaquinta,M. Multiple integrals in the calculus of variations and nonlinear elliptic systems; Annals of Mathematics Studies; Princeton University Press:Princeton, NJ, 1983; Vol. 105. [2] Gilbarg,D.;Trudinger,N.S. Elliptic partial differential equations of second order, 2nd Ed.; Grundlehren der Mathematischen Wissenschaften; Springer-Verlag:Berlin, 1983.; Vol. 224. [3] Kang,K.;Kim,S. On the H¨ older continuity of solutions of a certain system related to Maxwell’s equations. SIAM J. Math. Anal.in press. [4] Ladyˇ zenskaja,O.A.;Solonnikov,V.A.;Ural’ceva,N.N. Linear and quasilinear equations of parabolic type; Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, 1967; Vol. 23. [5] Landau,L.D.;Lifshitz,E.M. Electrodynamics of continuous media; Pergamon Press:New York, 1960. [6] Lieberman,G.M. Second order parabolic differential equations, World Scientific Publishing Co., Inc.: River Edge, NJ, 1996. ˇ ˇas,J.;Sver ´k,V. On regularity of solutions of nonlinear parabolic systems. Ann. Scuola [7] Nec a Norm. Sup. Pisa Cl. Sci. (4) 1991, 18 (1), 1–11. [8] Struwe,M. On the H¨ older continuity of bounded weak solutions of quasilinear parabolic systems Manuscripta Math. 1981, 35 (1-2), 125–145. [9] Yin,H.-M. Global solutions of Maxwell’s equations in an electromagnetic field with a temperature-dependent electrical conductivity. European J. Appl. Math. 1994, 5 (1), 57–64. [10] Yin,H.-M. Regularity of solutions to Maxwell’s system in quasi-stationary electromagnetic fields and applications. Comm. Partial Differential Equations 1997, 22 (7-8), 1029–1053. [11] Yin,H.-M. On Maxwell’s equations in an electromagnetic field with the temperature effect. SIAM J. Math. Anal. 1998, 29 (3), 637–651 (electronic). [12] Yin,H.-M. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Commun. Pure Appl. Anal. 2002, 1 (1), 127–134.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 E-mail address: [email protected] E-mail address: [email protected] Institute for Mathematics and its Applications, University of Minnesota E-mail address: [email protected]
Abstract. The goal of this paper is to establish H¨ older estimates for the solutions of a certain parabolic system related to Maxwell’s equations. Such an estimate is employed to get the local H¨ older continuity of the magnetic field arising from Maxwell’s equations in a quasi-stationary electromagnetic field, provided the resistivity of the material is continuous in time.
1. Introduction Let Ω be a domain in R3 and let Q = Ω × (0, T ) be a cylinder in R4 . Let A(x, t) be a 3 × 3 symmetric matrix such that there exists a number ν ∈ (0, 1) satisfying (1.1)
2
ν |ξ| ≤ hA(x, t)ξ, ξi
and
|A(x, t)| ≤ ν −1
∀(x, t) ∈ Q.
Here hξ, ηi denotes the usual inner product of vectors ξ, η ∈ R3 and |X| is the 2 Euclidean length of X ∈ RN , i.e., |X| is the sum of squares of each entry of X. In this paper we study the regularity of the solutions of the following system: ut + ∇×[A(x, t)∇×u] = 0 (1.2) in Q. ∇·u=0 Here, we denote ∇×u = curl u and ∇ · u = div u, where u = (u1 , u2 , u3 ) ∈ R3 . Recently, in [3], the elliptic case of this system has been studied by the first two authors. Without imposing any assumptions other than (1.1), they derived a priori H¨ older estimate of its weak solutions. Although a special type of A(x) was considered in the article [3], the main result [3, Theorem 2.1] remains valid as long as A(x) satisfies (1.1). We would like to mention that independently, Yin obtained a similar result in [12]. In this context, it is an interesting question to ask whether or not the weak solutions of the system (1.2) are locally H¨older continuous when the coefficients A(x, t) are assumed to be only measurable. Our main result states that weak solutions of the system (1.2) are locally H¨older continuous in the case that A(x, t) is uniformly continuous in time or A(x, t) is of the form a(t)B(x) where a(t) is a scalar function and B(x) is a symmetric matrix (see Theorem 3.2 and Theorem 3.3 below). The main idea is to use the elliptic result of [3], as well as standard perturbation techniques and the structure of the system (1.2). As an application of the main result, we consider the following system arising from Maxwell’s equations in a quasi-stationary electro-magnetic field where the The first author was supported in part by NSF Grant No. DMS-9877055. The second author was supported in part by NSF Grant No. DMS-9971052. 1
2
KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
displacement of the electric current is assumed to be time independent: H t + ∇×[%(x, t)∇×H] = 0 in Q. ∇·H =0 Here, the vector H represents the strength of the magnetic field and % the electrical resistivity of the material (see [5], [10]). Using our result from Section 3, one finds that if % is assumed to be continuous in time, then H is locally H¨older continuous. Another application is the following coupled system introduced in [9]: H t + ∇×[%(u)∇×H] = 0 ∇·H =0 (1.3) in Q = Ω × (0, T ), 2 ut − ∆u = %(u) |∇×H| where H and u are the magnetic field and the temperature of the material, respectively, and %(u) is the electrical resistivity of the material. In [10] Yin proved the H¨ older continuity of (H, u) under the assumption that %(u) is continuous. Using our result of Section 3, the same conclusion holds, only assuming the continuity of %(u) in time. Finally we consider the Stokes system with measurable coefficients: ut − A(x, t)∆u + ∇p = f in Q = R3 × (0, ∞). div u = 0 With the same assumption on A(x, t) as in Section 3, we prove that u is locally H¨ older continuous. This paper is organized as follows: In Section 2, we introduce the notations and recall some known results used in our proofs. In Section 3, we study the system (1.2) and prove our main result. In Section 4, we study some parabolic systems related to Maxwell’s equations and the Stokes system with measurable coefficients, as applications. 2. Notations and Preliminaries In this section, we introduce the notations which will be used throughout this article and also recall some well-known facts. Let us begin with the notations. • z0 = (x0 , t0 ) denotes an arbitrary point in Rn+1 , where x0 ∈ Rn and t0 ∈ (−∞, ∞). • Br = Br (x0 ) = {x ∈ Rn : |x − x0 | < r}. • Qr = Qr (z0 ) = (x, t) ∈ Rn+1 : |x − x0 | < r, −r2 < t − t0 < 0 . • Qr,t = Qr,t (z0 ) = {(x, t) ∈ Qr (z0 )}; i.e., Qr,t (z0 ) = Br (x0 ) × {t} if t ∈ (t0 − r2 , t0 ) and Qr,t (z0 ) = ∅ otherwise. • For Q0 ⊂ Q, ∂p Q0 is the parabolic boundary of Q0 . • Q0 b Q means RQ0 is compact and Q0 ∪ ∂p Q0 ⊂ Q.R R R • We denote by S f Rthe average of f on S; i.e., S f = S f / S 1 and we denote fr = fz0 ,r = Qr (z0 ) f . • For Ω ⊂ Rn and q > 1, Lq (Ω) denotes the Banach space of measurable functions with the following norms: Z 1/q q kukLq (Ω) = |u(x)| dx and kukL∞ (Ω) = ess sup |u| . Ω
Ω
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
3
• Let Q = Ω × (a, b). For 1 ≤ q, r ≤ ∞, Lq,r (Q) denotes the Banach space of all measurable functions with the finite norm r/q !1/r Z b Z q kukq,r; Q = kukLq,r (Q) = |u(x, t)| dx dt . a
q,q
Ω
q
• L (Q) will be denoted by L (Q) and the norm k · kLq,q (Q) by k · kLq (Q) . • For 1 ≤ q ≤ ∞, W k,q (Ω) denotes the usual Sobolev space; i.e., W k,q (Ω) = {u : Dα u ∈ Lq (Ω), 0 ≤ |α| ≤ k}, and W0k,q (Ω) denotes the completion of C0∞ (Ω) in W k,q (Ω). • For 1 ≤ q, r ≤ ∞, Lr (a, b); W k,q (Ω) denotes the Banach space of all measurable functions with the finite norm !1/r Z b r kukLr ((a,b);W k,q (Ω)) = ku(·, t)kW k,q (Ω) dt . a
• C α,α/2 (Q) denotes the Banach space of functions that are H¨older continuous with the exponent α ∈ (0, 1), and |u(z) − u(z 0 )| , d(z, z 0 )α z6=z 0 ∈Q
[u]α,α/2; Q = [u]C α,α/2 (Q) = sup
1/2
where d(·, ·) is the parabolic metric; i.e., d(z, z 0 ) = |x − x0 | + |t − t0 | . α,α/2 • u ∈ Lp,q ) means u ∈ Lp,q (Q0 ) (resp., u ∈ C α,α/2 (Q0 )) for loc (resp., u ∈ Cloc 0 0 all Q = Ω × (a, b) b Q. • The Morrey space M 2,µ (Q) is defined to be the set of all functions u ∈ L2loc (Q) with the finite norm !1/2 Z kukM 2,µ (Q) =
sup
2
ρ−µ
Qρ (z)⊂Q
|u|
.
Qρ (z)
• We denote by N = N (α, β, . . .) a constant depending on the prescribed quantities α, β, . . .. The following lemma is Campanato’s integral characterization of H¨older continuous functions (see e.g. [6, Lemma 4.3, page 50]). Lemma 2.1. Let fR ∈ L2 (Q2R (z0 )) and suppose there are positive constants α ≤ 1 2 and H such that Qr (z) |f − fz,r | ≤ H 2 rn+2+2α for any z ∈ QR (z0 ) and any r ∈ (0, R). Then f is H¨ older continuous with the exponent α in QR (z0 ) and [f ]α,α/2;QR (z0 ) ≤ N (n, α)H. The following lemma can be found in Giaquinta [1, Lemma 2.1, page 86]. Lemma 2.2. Let φ(t) be a nonnegative and nondecreasing function. Suppose that i h ρ α + ε φ(r) + Brβ φ(ρ) ≤ A r for all ρ < r ≤ R, with A, α, β nonnegative constants, β < α. Then there exists a constant ε0 = ε0 (A, α, β) such that if ε < ε0 , for all ρ < r ≤ R we have ρ β β φ(ρ) ≤ c φ(r) + Bρ r where c is a constant depending on α, β, A.
4
KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
3. Main result This section deals with the linear theory and a priori estimates. To avoid the technicalities, the coefficient A(x, t) will be assumed to be smooth as in [3]. Nonetheless, the constant appearing in the a priori estimates will not depend on the extra smoothness of A(x, t). Also, by a solution, we always mean a smooth solution unless otherwise stated. Indeed, the energy inequality below indicates that our system (1.2) is strongly parabolic. Hence, the general theory on the parabolic systems of divergence type can be applied, and by requiring A(x, t) to be smooth, we easily see that the weak solutions of (1.2) are actually smooth (see, e.g. [4]). Lemma 3.1 (Energy inequality). Let u be a solution of (1.2). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then the following estimate holds: Z Z Z N (ν, λ) 2 2 2 sup |u(·, s)| + |∇u| ≤ |u| . 2 2 R t0 −R ≤s≤t0 BR QR QλR Proof. Assume, for simplicity, that z0 = (0, 0). Let η be a cut-off function which vanishes near ∂p QλR . Using η 2 u as a test function, we find (see e.g. [4] or [6]) Z Z Z 2 2 2 2 2 2 sup η |u(·, s)| + η |∇×u| ≤ N (ν) |ηt | + |∇η| |u| . −R2 ≤s≤0
Ω
Q
Q
On the other hand, from the vector identity ∇×∇×u = −∆u + ∇(∇ · u)
(3.1)
and the equation ∇ · u = 0, we find Z Z Z 2 2 2 2 2 2 η |∇u| ≤ N η |∇×u| + |∇η| |u| . Q
Q
Q
Combining the above inequalities and choosing a suitable η, we finish the proof. Next lemma states that if A(x, t) does not depend on time; i.e., A = A(x), then any solution u of (1.2) satisfies ut ∈ L2loc (Q) and ∇u ∈ L2,∞ loc (Q). Lemma 3.2. Let u be a solution of (1.2) with A = A(x). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then, we have Z Z Z N (ν, λ) 2 2 2 |ut | + sup |∇u(·, s)| ≤ |u| . R4 t0 −R2 ≤s≤t0 BR QR QλR Proof. As before, we assume z0 = (0, 0). Let η be a cut-off function vanishing near ∂p Q√λR . Using η 2 ut as a test function, we find (see, e.g. [4]) Z Z Z 2 2 2 2 η 2 |ut | + sup η 2 |∇×u(·, s)| ≤ N (ν) (|ηt | + |∇η| ) |∇u| . −R2 ≤s≤0
Q
Ω
Q
Also, from (3.1) we find that Z Z Z 2 2 2 2 η 2 |∇u(·, s)| ≤ N η 2 |∇×u(·, s)| + |∇η| |u(·, s)| . Ω
Ω
Ω
Combining the√above inequalities and applying Lemma 3.1 with QR and λ replaced by Q√λR and λ, respectively, we complete the proof. Next lemma says that if A = A(x), then ut belongs to L2,∞ loc (Q). This is where the result of the elliptic case proved in [3] is applied.
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
5
Lemma 3.3. Let u be a solution of (1.2) with A = A(x). Let R > 0 be such that QλR := QλR (z0 ) ⊂ Q for some λ > 1. Then, the following estimate holds: Z Z Z N (ν, λ) 2 2 2 sup |ut (·, s)| + |∇ut | ≤ |u| . 6 2 R t0 −R ≤s≤t0 BR QR Q Proof. As before, we assume z0 = (0, 0). Taking the derivatives with respect to time in (1.2), we get utt + ∇×(A∇×ut ) = 0,
∇ · ut = 0.
Applying Lemma 3.1 to ut , and then using Lemma 3.2 we find Z Z Z Z N (ν, λ) N (ν, λ) 2 2 2 2 sup |ut (·, s)| + |∇ut | ≤ |ut | ≤ |u| . 2 6 2 R R −R ≤s≤0 BR QR Q√λR QλR This completes the proof.
The following lemma is adapted from Struwe [8]. One can easily verify it by following the proof of Lemma 3 in [8] line by line. Lemma 3.4. Let u be a solution of (1.2). Let R > 0 be such that QλR = QλR (z0 ) ⊂ Q for some λ > 1. Then, there exist a constant N = N (ν, λ) such that Z Z 2 2 (3.2) |u − uR | ≤ N R2 |∇u| . QR
QλR
The next theorem is our first main result. The idea is to treat ut as an inhomogeneous term and apply the elliptic result of [3]. This kind of approach is found, for example, in [7]. Theorem 3.1. Let u be a solution of (1.2) with A = A(x). Then, u is locally H¨ older continuous in Q. Moreover, the following estimate holds: for any R > 0 such that Q6R = Q6R (z0 ) ⊂ Q, [u]C α,α/2 (QR ) ≤ N (ν)R−(5+2α) kukL2 (Q6R ) , where α = α(ν) ∈ (0, 1/2]. Proof. We assume that R = 1 and z0 = (0, 0). The general case is recovered by a simple coordinate change (x, t) 7→ ((x − x0 )/R, (t − t0 )/R2 ). From Lemma 3.3, we know that ut ∈ L2,∞ (Q5 ). Also, we find that ∇ · ut = 0 from the vector identity ∇ · ∇×F = 0. Hence, we can rewrite (1.2) as ∇×(A(x)∇×u) = −ut
(3.3)
and apply the elliptic result [3, Theorem 2.1] to find that for any t ∈ (−42 , 0), we have (3.4) [u(·, t)]C α (Q4,t ) ≤ N (ν) ku(·, t)kL2 (Q5,t ) + kut (·, t)kL2 (Q5,t ) , where α = α(ν) ∈ (0, 1/2] (see Remark 3.1 below). From Lemma 3.1 and Lemma 3.3, the right hand side of (3.4) is uniformly bounded for t ∈ (−42 , 0) hence (3.5)
[u(·, t)]C α (Q4,t ) ≤ N (ν) kukL2 (Q6 )
∀t ∈ (−42 , 0).
Fix z = (x, t) ∈ Q1 and r ≤ 1. From Lemma 3.4, we have Z Z 2 2 (3.6) |u − uz,r | ≤ N r2 |∇u| . Qr (z)
Q2r (z)
6
KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
From (3.3), the following Caccioppoli type inequality holds (see [3, Lemma 4.4]). ! Z Z 1 2 2 2 |u − u(t)| + kut (·, t)kL6/5 (Q3r,t (z)) , |∇u| ≤ N r2 Q3r,t (z) Q2r,t (z) R where u(t) = Q3r,t (z) u(·, t). Note that, from (3.5), Z 1 2 2 |u − u(t)| ≤ N r1+2α [u(·, t)]2C α (Q4,t ) ≤ N r1+2α kukL2 (Q6 ) . r2 Q3r,t (z) Also, from H¨ older’s inequality and Lemma 3.3 2
2
2
kut (·, t)kL6/5 (Q3r,t (z)) ≤ N r2 kut (·, t)kL2 (Q3r,t (z)) ≤ N r2 kukL2 (Q6 ) . Therefore, we have Z 2 |∇u| Q2r (z)
=
Z
t
t−4r 2
Z
2
|∇u(y, s)| dyds
Q2r,t (z)
2 2 ≤ N r2 r1+2α + r2 kukL2 (Q6 ) ≤ N r3+2α kukL2 (Q6 ) , where we used α ≤ 1/2 in the last step. Hence, using (3.6) we conclude Z 2 2 |u − uz,r | ≤ N (ν)r2α kukL2 (Q6 ) , ∀r ≤ 1, ∀z ∈ Q1 . Qr (z)
Finally, from Lemma 2.1, we get [u]α,α/2,Q1 ≤ N (ν) kukL2 (Q6 ) . This completes the proof. Remark 3.1. In [3, Theorem 2.1], A(x) is assumed to be of the form a(x)I. However, the proof works for general A(x) as long as A(x) satisfies (1.1). Also, one can find in the proof that α ≤ γ, where γ = 2 − 3/2 = 1/2 in our case. Definition 3.1. Let f : Q → RN be measurable. For all r > 0 and z = (x, t) such that Qr (z) ⊂ Q, we define ωf (r; z) := ess sup(y,s)∈Qr (z) |f (y, t) − f (y, s)| and (3.7)
ωf (r) := sup {ωf (r; z) : ∀z ∈ Q such that Qr (z) ⊂ Q} .
Lemma 3.5. Let u be a solution of (1.2). Suppose there is a fixed τ ∈ (0, 1) such that for all R > 0 satisfying QR ⊂ Q, [u]C α,α/2 (Qτ R ) ≤ N R−(5+2α)/2 kukL2 (QR ) .
(3.8)
Then for all 0 < ρ ≤ r ≤ R, we have the following estimate for ∇u: Z ρ 3+2α Z 2 2 |∇u| ≤ N (τ ) (3.9) |∇u| . r Qr Qρ Proof. We may assume ρ < (τ /4)r, for (3.9) is obvious if ρ ≥ (τ /4)r. We denote [u]α,r := [u]C α,α/2 (Qr ) . Applying the energy inequality to u − u2ρ , which is also a solution of (1.2), we find (recall 2ρ ≤ (τ /2)r) Z Z 2 2 −2 (3.10) |∇u| ≤ N ρ |u − u2ρ | ≤ N ρ3+2α [u]2α,(τ /2)r . Qρ
Q2ρ
From (3.8) applied to v = u − ur/2 , and by Lemma 3.4, Z Z u − ur/2 2 ≤ N r−3−2α [u]2α,(τ /2)r = [v]2α,(τ /2)r ≤ N r−5−2α Qr/2
Putting the above inequalities together, we complete the proof.
2
|∇u| .
Qr
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
7
Theorem 3.2. Let u be a solution of (1.2) with A(x, t) satisfying: (A) limρ→0 ωA (ρ) = 0, where ωA is defined as in Definition 3.1. Then, u is locally H¨ older continuous in Q. More precisely, let β ∈ (0, α), where α is the H¨ older exponent in Theorem 3.1, and let Q0 b Q. Then, there is Q00 such that Q0 ⊂ Q00 b Q and [u]C β,β/2 (Q0 ) ≤ N (ν, β, ωA , Q0 , Q) kukL2 (Q00 ) . Proof. Let R0 be a fixed number which will be specified later. Fix z0 ∈ Q0 and let R > 0 be such that 2R ≤ R0 and Q4R = Q4R (z0 ) ⊂ Q. We will show that [u]C β,β/2 (QR ) ≤ N (ν, β, R) kukL2 (Q4R ) . Then, the theorem will follow from a standard covering argument. Fix z = (x, t) ∈ QR and let 0 < ρ < r ≤ R. For any z 0 = (y, t) ∈ QR , denote A0 (y) = A(y, t) and let v be a solution of the system v t + ∇×[A0 ∇×v] = 0,
∇ · v = 0 in Q2r (z)
with the boundary condition v = u on ∂p Q2r (z). Let α = α(ν) be the H¨ older exponentR from Theorem 3.1. From (3.9) applied to R 2 2 v, we have Q2ρ (z) |∇v| ≤ N (ρ/r)3+2α Q2r (z) |∇v| , and hence, # " Z Z ρ 3+2α Z 2 2 2 |∇u| + |∇(u − v)| . |∇u| ≤ N r Q2r (z) Q2r (z) Q2ρ (z) Using the equations satisfied by u and v, the function w = u − v satisfies wt + ∇×[A0 ∇×w] = ∇×[(A0 − A)∇×u],
∇ · w = 0 in Q2r (z)
and w = 0 on ∂p Q2r (z). By using w itself as a test function to the above equation R R 2 2 we find Q2r (z) |∇w| ≤ N ωA (2r) Q2r (z) |∇u| (recall ∇ · w = 0) and hence # " Z Z ρ 3+2α Z 2 2 2 |∇u| ≤ N |∇u| + ωA (2r) |∇u| . r Q2ρ (z) Q2r (z) Q2r (z) Now, choose R0 such that ωA (R0 ) < ε0 , where ε0 is as in Lemma 2.2. From Lemma 3.4, Lemma 2.2, and Lemma 3.1, we get Z Z 2 2 |u − uz,r | ≤ N (ν)r2 |∇u| Qr (z) Q2r (z) Z Z 2 2 ≤ N (ν, β, R)r5+2β |∇u| ≤ N (ν, β, R)r5+2β |u| . Q2R
Therefore, the theorem follows from Lemma 2.1.
Q4R
Let A(x, t) ≡ a(t)B(x), where a(t) is a positive, bounded scalar function and B is a symmetric matrix satisfying (1.1). The hypothesis (A) in Theorem 3.2 is not satisfied in this case unless a(t) is assumed to be continuous. However, in such a case, it turns out that we don’t need to impose any continuity assumptions on a(t) to get the H¨ older estimate for u, as we will show next. Theorem 3.3. Let u be a solution of (1.2) with A(x, t) = a(t)B(x), where a(t) is a scalar function such that ν ≤ a(t) ≤ ν −1 and B(x) is a symmetric matrix satisfying (1.1). Then, for any R > 0 such that Q7R = Q7R (x0 ) ⊂ Q, u is H¨ older continuous in QR and (3.11)
[u]C α,α/2 (QR ) ≤ N (ν)R−(5+2α)/2 kukL2 (Q7R ) ,
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
where α = α(ν) ∈ (0, 1/2]. Proof. We may assume z0 = (0, 0) and R = 1 as before. From the assumption that A = a(t)B(x), we see that (1.2) becomes (3.12)
ut + a(t)∇×[B(x)∇×u] = 0,
∇ · u = 0.
2 −1
Using η a But as a test function in (3.12) and proceeding as in the proof of Lemma 3.2, we obtain the estimate Z Z 2 2 2 (3.13) |ut | + sup |∇u(·, s)| ≤ N (ν) kukL2 (Q7 ) . −62 ≤s≤0
Q6
B6
2
In particular, we have ut ∈ L (Q6 ). Next, by taking the curl in the equation (1.2) and denoting w = ∇×u, we get (3.14)
wt + a∇×[∇×Bw] = 0. 2 −1
Using η a (3.15)
Z
Bwt as a test function in (3.14) and proceeding as above, we find Z Z 2 2 2 |wt | + sup |∇×Bw| ≤ N (ν) |∇×Bw| . −52 ≤s≤0
Q5
B5
Q6
−1
RSince ∇×Bw2 = −a ut from (3.12), the right hand side of (3.15) is estimated as |∇×Bw| ≤ N (ν) kukL2 (Q7 ) from (3.13). Hence, it follows that Q6 2
kut kL2,∞ (Q5 ) ≤ N (ν) k∇×BwkL2,∞ (Q5 ) ≤ N (ν) kukL2 (Q7 ) . The rest of proof goes exactly as in that of Theorem 3.1, and therefore, the details are omitted. Corollary 3.1. Let β ∈ (0, α), where α is the H¨ older exponent in Theorem 3.3. Suppose u is a solution of (1.2) with A(x, t) satisfying the following assumption: there exists A0 (x, t) = a(t)B(x) such that (A0 ) |A(z) − A0 (z)| ≤ ε for all z ∈ Q2R ⊂ Q, where a(t) and B(x) satisfy the assumptions in Theorem 3.3. Then, there exists ε0 = ε0 (ν, β) such that if ε < ε0 , u is H¨ older continuous in QR and [u]C β,β/2 (QR ) ≤ N (ν, β, R) kukL2 (Q2R ) . Proof. From Theorem 3.3 and Lemma 3.5, we see that the estimate (3.9) holds. Hence, the proof of Theorem 3.2 works here. Remark 3.2. We may also consider the system (1.2) with an inhomogeneous term: ut + ∇×[A(x, t)∇×u] = ∇×f (3.16) in Q. ∇·u=0 Let β ∈ (0, α), where α is the H¨older exponent in Theorem 3.1. Suppose f ∈ M 2,3+2β (Q) and A(x, t) satisfies (A) or (A0 ). Let u be a solution of (3.16). It can be easily verified, by modifying the proof of Theorem 3.2, that u is H¨older continuous in Q0 b Q and that it satisfies the following estimate: [u]C β,β/2 (Q0 ) ≤ N (ν, β, Q0 , Q) kukL2 (Q) + kf kM 2,3+2β (Q) .
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
9
4. Applications We first consider a certain system arising in a Maxwell’s equations. Let a conductive material occupy a bounded domain Ω ⊂ R3 . Let E and H be the electric and the magnetic fields in Ω. Under certain assumptions, Yin derived the following mathematical model (see [9], [11]): (E)t + σE = ∇×H, (µH)t + ∇×E = 0, (4.1) in Q, div(µH) = 0, where , µ, and σ are the electric permittivity, the magnetic permeability and the electric conductivity of the material, respectively. If the electrical displacement D = E is negligible and µ is a constant, the above system can be reduced to H t + ∇×[%(x, t)∇×H] = 0, (4.2) in Q, div H = 0, where % is the resistivity of the material (see [9], [10]). The above system is the special case of the system (1.2), where A = %(x, t)I and I is the identity matrix. Therefore, our result implies that H is locally H¨older continuous in Q provided that %(x, t)I satisfies the assumption (A) or (A0 ) of the previous section. Theorem 4.1. Let H be a weak solution of (4.2). Suppose that %I satisfies either (A) or (A0 ). Then H is locally H¨ older continuous. In fact, the following estimate holds: for R > 0 such that Q2R = Q2R (z0 ) ⊂ Q, there exists α ∈ (0, 1/2) such that [H]C α,α/2 (QR ) ≤ N kHkL2 (Q) . Next, we consider the case when temperature affects the electrical resistance. By taking the temperature effect into consideration, Yin derived the following mathematical model (see Yin [9], [10]): H t + ∇×[%(u)∇×H] = 0 ∇·H =0 (4.3) in Q = Ω × (0, T ), 2 ut − ∆u = %(u) |∇×H| where H and u are the magnetic field and the temperature of the material, respectively, and %(u) is the electrical resistivity of the material. Let us assume that (4.4)
lim ω%(u) (r) = 0,
r→0
where ω%(u) (r) is defined as in Definition 3.1. The assumption (4.4) above is an analogue of the assumption (A) in the previous section. Roughly speaking, this assumption is to say that the resistivity %(u) varies continuously in time, uniformly throughout the material Ω. For instance, the assumption (4.4) is satisfied if the scalar function % is Lipschitz continuous and the temperature u is continuous in time variable so that limr→0 ωu (r) = 0. We will show that, under the assumption (4.4), a pair of weak solution (H, u) of the system (4.3) is locally H¨older continuous. We remark that in [10], the continuity of %(u) in both spatial and time directions is assumed to get the same result. Theorem 4.2. Let (H, u) be a pair of weak solutions of (4.3). Suppose that %(u) satisfies (4.4). Then (H, u) is locally H¨ older continuous.
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT
The following lemmas will be used in the proof of Theorem 4.2 above. The proof of Lemma 4.1 below is standard (see, e.g. [8, Lemma 3]) and it will be omitted. Lemma 4.1. Let u be a solution of ut − ∆u = ∇ · f + g
(4.5)
in
Q.
Then the following estimate holds: for R > 0 such that Q2R ⊂ Q Z Z Z Z 2 2 2 2 |u − uR | ≤ N R2 |∇u| + R2 |f | + R4 |g| . QR
Q2R
Q2R
Q2R
Lemma 4.2. Let u be a smooth solution of (4.5) in Q and let R > 0 such that Q2R ⊂ Q. Assume that f ∈ M 2,3+2α (Q2R ) and g ∈ M 2,1+2α (Q2R ). Then, ∇u ∈ M 2,3+2α (QR ) and 2 2 2 2 k∇ukM 2,3+2α (QR ) ≤ N (R) kukL2 (Q2R ) + kf kM 2,3+2α (Q2R ) + kgkM 2,1+2α (Q2R ) . Sketch of proof. Fix z ∈ QR and r ∈ (0, R). Let v be the solution of vt − ∆v = 0 in Qr (z), v = u on ∂p Qr (z). R 2 2 Then, Qρ (z) |∇v| ≤ N (ρ/r)5 Qr (z) |∇v| for all ρ ≤ r. Let w := u − v. Note that w satisfies wt − ∆w = ∇ · f + g in Qr (z) and w = 0 on ∂p Qr (z). 2 2 Denote F = kf kM 2,3+2α (Q2R ) and G = kgkM 2,1+2α (Q2R ) . One can easily check R 2 that Qr (z) |∇w| ≤ N (F + G)r3+2α . Then, using Lemma 2.2, one may conclude R R 2 2 that Qr (z) |∇u| ≤ N [(r/R)3+2α QR |∇u| + (F + G)r3+2α ]. On the other hand, R R R R 2 2 2 2 from the energy inequality, QR |∇u| ≤ N (R−2 Q |u| + Q2R |f | + R2 Q2R |g| ). R 2 2 Therefore, we derived r−3−2α Qr (z) |∇u| ≤ N (R){kuk2;Q2R + F + G}. R
Now we are ready to prove Theorem 4.2. Proof of Theorem 4.2. We may assume kHk2 = kHk2;Q < ∞. Fix z0 ∈ Q and let R > 0 be such that Q8R = Q8R (z0 ) b Q. It is enough to show that (H, u) is H¨ older continuous in QR . From Theorem 3.2, there is 0 < α < 1/2 such that [H]C α,α/2 (Q8R ) ≤ N kHk2 , where N is a constant depending on the given data. Thus, using the triangle inequality, we find that H is locally bounded and kHkL∞ (Q4R ) ≤ N kHk2 .
(4.6)
It remains to show that u is H¨older continuous in QR . Using the following vector identity ∇ · (F × G) = (∇×F ) · G − F · (∇×G) and (4.3), we observe that 2
%(u) |∇×H| = ∇ · [H × (%(u)∇×H)] + H · H t , and thus the last equation of (4.3) can be rewritten as (4.7)
ut − ∆u = ∇ · [H × (%(u)∇×H)] + H · H t =: ∇ · f + g.
Fix z ∈ QR and r ∈ (0, R). Using (4.6) and proceeding as in (3.10) in the proof of Lemma 3.5 Z Z 2 2 2 4 |f | ≤ N kHk2 (4.8) |∇H| ≤ N r3+2α kHk2 . Q2r (z)
Q2r (z)
¨ ON THE HOLDER CONTINUITY OF SOLUTIONS OF A CERTAIN SYSTEM
11
Similarly, using (4.6) and applying Lemma 3.3 to H (recall 2α ≤ 1), Z Z t Z 2 2 2 4 (4.9) |g| ≤ N kHk2 |H t | ≤ N r1+2α kHk2 . t−4r 2
Q2r (z)
Q2r (z)
Now, we apply Lemma 4.1 to (4.7) in Q2r (z) using (4.8) and (4.9) Z Z 2 2 4 |u − uz,r | ≤ N r2 |∇u| + N r5+2α kHk2 . Qr (z)
Q2r (z)
Hence, taking the average Z Z 2 (4.10) |u − uz,r | ≤ N r−3 Qr (z)
Q2r (z)
2
4
|∇u| + N r2α kHk2 .
Applying Lemma 4.2 to (4.7) and using (4.8) and (4.9), we conclude Z 2 2 4 −3−2α (4.11) r |∇u| ≤ N kukL2 (Q4R ) + kHk2 . Q2r (z)
Finally, by combining (4.10) and (4.11), we have Z 2 2 2 |u − uz,r | ≤ N r2α kukL2 (Q4R ) + kHkL2 (Q) . Qr (z)
Therefore, [u]α,α/2;QR < ∞. This completes the proof.
Next we consider the time-dependent Stokes system with measurable coefficients in diffusive term ut − A(x, t)∆u + ∇p = f (4.12) in Q ≡ R3 × (0, ∞). div u = 0 Here u and p are the velocity field and the pressure associated with u and f is a given external force. A(x, t) is a 3 × 3 symmetric matrix satisfying the ellipticity condition (1.1). The following theorem is the last application of our results from Section 3. Theorem 4.3. Let f ∈ M 2,3+2β (Q), where 0 < β < α and α is the H¨ older exponent in Theorem 3.1. Suppose ut , ∇p ∈ L2 (Q) and u ∈ L2 (0, ∞); W 2,2 (R3 ) such that u solves the system (4.12). Suppose that A(x, t) satisfies (A) or (A0 ). Then ∇u is locally H¨ older continuous and satisfies the following estimate: for R > 0 such that QR = QR (z0 ) ⊂ Q, we have [∇u]C β,β/2 (QR ) ≤ N k∇ukL2 (Q) + kf kM 2,3+2β (Q) , where N = N (ν, R). Proof. By taking the curl in (4.12), we have the following equations: wt + ∇ × (A(x, t)∇×w) = ∇ × f ,
w = ∇×u
in Q,
where we used the identity ∆u = −∇×∇×u + ∇(∇ · u). Then it is easy to see that w is locally H¨ older continuous in Q and the following estimate holds: [w]C β,β/2 (Q2R ) ≤ N (kwkL2 (Q4R ) + kf kM 2,3+2β (Q4R ) ), where N = N (ν, R). Using the Biot-Savart law, u can be recovered in terms of w as follows: Z (x − y) × w(y, t) 1 dy. u(x, t) = − 3 4π R3 |x − y|
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KYUNGKEUN KANG, SEICK KIM, AND AURELIA MINUT β,β/2
It is easy to see that ∇u ∈ Cloc [∇u]C β,β/2 (QR )
(Q) and the following estimate holds: ≤ N [w]C β,β/2 (Q2R ) ≤ N k∇ukL2 (Q) + kf kM 2,3+2β (Q) .
This completes the proof.
References
[1] Giaquinta,M. Multiple integrals in the calculus of variations and nonlinear elliptic systems; Annals of Mathematics Studies; Princeton University Press:Princeton, NJ, 1983; Vol. 105. [2] Gilbarg,D.;Trudinger,N.S. Elliptic partial differential equations of second order, 2nd Ed.; Grundlehren der Mathematischen Wissenschaften; Springer-Verlag:Berlin, 1983.; Vol. 224. [3] Kang,K.;Kim,S. On the H¨ older continuity of solutions of a certain system related to Maxwell’s equations. SIAM J. Math. Anal.in press. [4] Ladyˇ zenskaja,O.A.;Solonnikov,V.A.;Ural’ceva,N.N. Linear and quasilinear equations of parabolic type; Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, 1967; Vol. 23. [5] Landau,L.D.;Lifshitz,E.M. Electrodynamics of continuous media; Pergamon Press:New York, 1960. [6] Lieberman,G.M. Second order parabolic differential equations, World Scientific Publishing Co., Inc.: River Edge, NJ, 1996. ˇ ˇas,J.;Sver ´k,V. On regularity of solutions of nonlinear parabolic systems. Ann. Scuola [7] Nec a Norm. Sup. Pisa Cl. Sci. (4) 1991, 18 (1), 1–11. [8] Struwe,M. On the H¨ older continuity of bounded weak solutions of quasilinear parabolic systems Manuscripta Math. 1981, 35 (1-2), 125–145. [9] Yin,H.-M. Global solutions of Maxwell’s equations in an electromagnetic field with a temperature-dependent electrical conductivity. European J. Appl. Math. 1994, 5 (1), 57–64. [10] Yin,H.-M. Regularity of solutions to Maxwell’s system in quasi-stationary electromagnetic fields and applications. Comm. Partial Differential Equations 1997, 22 (7-8), 1029–1053. [11] Yin,H.-M. On Maxwell’s equations in an electromagnetic field with the temperature effect. SIAM J. Math. Anal. 1998, 29 (3), 637–651 (electronic). [12] Yin,H.-M. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Commun. Pure Appl. Anal. 2002, 1 (1), 127–134.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 E-mail address: [email protected] E-mail address: [email protected] Institute for Mathematics and its Applications, University of Minnesota E-mail address: [email protected]
