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Topics of Regularity in PDE workshop

Solvability problem for strong nonlinear nondiagonal parabolic systems 1 A. A. ARKHIPOVA St.Petersburg State University, Department of Mathematics and Mechanics, Petrodvorets, Bibliotechnaya pl.2, 198504 St.Petersburg, RUSSIA e-mail: [email protected]

Let Ω be a bounded domain in Rn , n ≥ 2, with sufficiently smooth boundary. For a fixed T > 0 and Q = Ω × (0, T ) we consider a solution u : Q → RN , u = (u1 , . . . , uN ), N > 1, of the parabolic system (1) ukt −

d k a (z, u, ux ) + bk (z, u, ux ) = 0, dxα α

z = (x, t) ∈ Q,

k = 1, . . . , N.

We define the set D = Q × RN × RnN and assume that k k≤N are smooth enough on D; a) the function a = {akα }k≤N α≤n and b = {b }

b) for a fixed q > 1 a(·, ·, p) ∼ |p|q−1 , b(·, ·, p) ∼ |p|q , |p|  1; all derivarives of a and b, we need, have the natural growth in the gradient;

k

α α,β≤n c) nondiagonal principal matrix { ∂a } satisfies on D the assump∂plβ k,l≤N tions:

∂akα (z, u, p) k l ξα ξβ ≥ ν(l + |p|)q−2 |ξ|2 , l ∂pβ

|

∂a(. . . ) | ≤ µ(l + |p|)q−2 , ∂p

∀ξ ∈ RnN ;

d) strong-nonlinear term b satisfies the condition |b(z, u, p)| ≤ b0 (l + |p|)q ,

(z, u, p) ∈ D.

Here ν, µ, b0 = const > 0, l = const ≥ 0 (l 6= 0 if q 6= 2). 1

Supported by the Russian Foundation for Fundamental Studies (grant no.99-01 00684).

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A. A. Arkhipova We treat the solvability question for the Cauchy-Dirichlet problem (2)

u|∂ 0 Q = ϕ,

where ∂ 0 Q is the parabolic boundary of Q, and ϕ is a given smooth function. Note, that one can not expect classical solvability of (1), (2) under assumptions a)–d). Indeed, counterexamples of the regularity of solutions were constructed for quasilinear nondiagonal parabolic system (b = 0, q = 2, n > 2) (see [1]). From the other hand, let us consider heat flows of harmonic maps problem. In the model situation, u : Q → M ⊂ RN (M is a compact manifold) satisfies the system (1), where the main operator of the system is the heat operator, function b 6= 0 satisfies d) and has a special structure. It is known that in this problem singularities may appear in time inside of Q (see [2]). As we see, there exist two reasons of nonsmoothness of global solution for (1), (2). In general, solvability problem for (1), (2) is open. The author studied this problem for a class of systems with variational structure. More exactly, we introduce a functional Z E[u] = f (x, u, ux ) dx, Ω

and denote by L = {Lk }k≤N , the Euler operator of E: Lk u = −

d f k (x, u, ux ) + fuk (x, u, ux ). dxα pα

In this case, system (1) is the gradient flow for E[u], and we study the following problem (3)

ukt − dxdα fpkα + fuk = 0, z ∈ Q, u|Γ = 0, u|t=0 = ϕ0 (x),

where Γ = ∂Ω × (0, T ). As an example, we introduce f (x, u, p) =< A(x, u)p, p > (l + |p|)q−2 , βα q > 1. Let A be a nondiagonal positive definite matrix, Aαβ kl = Alk . Norms kAk, kA0u k are bounded on Ω × RN . In this case akα = fpkα , bk = fuk satisfy all assumptions a)–d).

66

Topics of Regularity in PDE workshop Due to the variational structure, we have an apriori estimate for a solution

u: kut k22,Q + sup kux (·, t)kqq,Ω ≤ e0 ,

(4)

(0,T )

where e0 = const depends on the data only. The solvability problem was investigated by the author for the case of two spatial variables (n = 2). I. The case n = q = 2. Problem (3) was studied for quasilinear and nonlinear operators, under Dirichlet and Neumann conditions [4] – [6]. In all situations we prove the following result. For any fixed T > 0 there exists a global almost everywhere smooth solution u of (3) in Q. The singular set of u consists of at most finitely many points. Function u has finite norms (4) and it is a weak solution of (3) in the sence of distributions. To state this result used the fact that the ”normalized R we essencially 1 q |ux (x, t)| dx is a monotonic function of R > 0, when local energy” Rn−q BR (x0 )

n = q = 2. II. The case n = 2, q > 2. Note, that in this case an apriori estimate (5)

kukC α (Q,δq ) ≤ c(e0 ) with some α ∈ (0, 1)

follows from (4). Here δq (z 1 , z 2 ) = sup{|x1 − x2 |, |t1 − t2 |1/q } defines qparabolic metric in Rn+1 . Estimates of stronger norms of u can be derived from (4) and (5). As a result, it is not difficult to prove: For any T > 0 there exists a smooth in Q solution u of problem (3). III. The case n = 2, q ∈ (1, 2). In this situation we approximate our problem in the following way. For ∀ε ∈ (0, 1) we consider the problem  k vt + Lk (v) − ε4v k = 0 (6) v|Γ = 0, v|t=0 = ϕε , where ϕε satisfies the compatibility condition for the new operator and ϕε → ϕ, ε → 0 in a strong sence. We prove the existence of a global smooth solution uε for (6), when ε > 0 is fixed.

A. A. Arkhipova

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The limit function u for the sequence uε , ε → 0, is a solution of problem (3) in the following sence: There exists a function u ∈ L∞ ((0, T ); W01,q (Ω)), with ut ∈ L2 (Q); u is almost everywhere smooth solution of (3) in Q. Closed singular set Σ of u has dimq−H Σ ≤ 2 (in the sence of q-parabolic metric). Moreover, dimH Στ ≤ 2 − q, ∀τ > 0, where Στ = Σ ∩ {t = τ }.

References [1] Stara J., John O. Some (new) counterexamples of parabolic systems. Comment. Math. Univ. Carolinae, 36, 3 (1995) 503-510. [2] Chen Y.,Struve M. Existence and partial regularity results for the heatflow for harmonic maps. Math.Z.,201, (1989) 83-103. [3] Chang K.C. Heat flow and boundary value problem for harmonic maps. Ann. Inst.Henri Poincare, 6,5 (1989) 363-395. [4] Arkhipova A.A. Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables. J. Math. Sci.,92, 6 (1998) 4231-4255. [5] Arkhipova A.A. Local and global in time solvability of the CauchyDirichlet problem to a class of nonlinear nondiagonal parabolic systems. St. Petersburg Math. J., 11, 6 (2000) 989-1017. [6] Arkhipova A.A. Cauchy-Neumann problem for a class of nondiagonal parabolic systems with guadratic growth nonlinearites. I.On the continuability of smooth solutions. II.Local and global solvability results. Comment. Math. Univ.Carolinae, 41, 4 (2000) 693-718; 42,1 (2001) 53-76.