Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows
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Calculus of Variations
Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows Baisheng Yan1 Received: 3 January 2020 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper is concerned with nonuniqueness and instability of the initial-boundary value problem for certain general systems of nonlinear diffusion equations. We explore the diffusion problems using the convex integration framework of nonhomogeneous space-time partial differential inclusions. Under a non-degeneracy (openness) condition called Condition (OC), we establish some nonuniqueness and instability results concerning Lipschitz solutions for such diffusion systems. For parabolic systems, this Condition (OC) proves to be compatible with strong polyconvexity. As a result, we prove that the initial-boundary value problem for gradient flows of certain 2×2 strongly polyconvex functionals possesses weakly* convergent sequences of exact Lipschitz solutions whose weak* limits are not a weak solution. Such an instability result cannot be obtained from the corresponding elliptic system. Mathematics Subject Classification 35K40 · 35K51 · 35D30 · 35F50 · 49A20
1 Introduction 1.1 Initial-boundary value problem for general diffusion system Let m, n ≥ 1, T > 0, and let Ω be a bounded domain in Rn with Lipschitz boundary ∂Ω. Assume u 0 : Ω¯ → Rm is a given (initial-boundary) function and σ = (σki (A)) : Mm×n → Mm×n is a given (diffusion flux) function; here Mm×n denotes the space of real m × n matrices. We study the initial-boundary value problem of diffusion equation: ⎧ u t = div σ (Du) in ΩT = Ω × (0, T ), ⎪ ⎪ ⎨ (1.1) (x ∈ ∂Ω, 0 < t < T ), u(x, t) = u 0 (x) ⎪ ⎪ ⎩ u(x, 0) = u 0 (x) (x ∈ Ω),
Communicated by J. Ball.
B 1
Baisheng Yan [email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 0123456789().: V,-vol
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B. Yan
for function u = (u 1 , . . . , u m ), where each component u i is a function of (x, t) ∈ ΩT , i u t = (u 1t , . . . , u m t ) is the time-derivative of u, Du = (u xk ) is the spatial Jacobi matrix of u. When m ≥ 2, the diffusion equation u t = div σ (Du) in (1.1) is a system of m quasilinear partial differential equations: u it =
n (σki (Du))xk (i = 1, 2, . . . , m).
(1.2)
k=1
By a weak solution to problem (1.1), we mean a function u ∈ W 1,1 (ΩT ; Rm ) that satisfies the initial-boundary conditions in (1.1) in the sense of trace and satisfies (1.2) in the sense that n u i φt − σki (Du)φxk d xdt = 0 ∀φ ∈ Cc∞ (ΩT ), i = 1, . . . , m. ΩT
k=1
We say Eq. (1.2) is strongly parabolic if the condition σ (A + p ⊗ α) − σ (A), p ⊗ α ≥ ν| p|2 |α|2
(1.3)
holds for all A ∈ Mm×n , p ∈ Rm and α ∈ Rn , where ν > 0 is a constant; here A, B stands for the inner product in Mm×n ∼ = Rmn and p ⊗ α denotes the matrix ( pi αk ). If m, n ≥ 2, it is well-known that condition (1.3) is strictly weaker than the full monotonicity condition: σ (A + B) − σ (A), B ≥ ν|B|2 ∀A, B ∈ Mm×n .
(1.4)
Note that if
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