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Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-07-10 , DOI: 10.1007/s00526-020-01785-7
Baisheng Yan

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 \(2\times 2\) 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.



中文翻译:

多凸梯度流的扩散方程和Lipschitz解的凸积分

本文关注某些非线性扩散方程组的初边值问题的非唯一性和不稳定性。我们使用非齐次时空偏微分包含的凸积分框架探索扩散问题。在称为条件(OC)的非简并性(开放性)条件下,我们建立了有关此类扩散系统的Lipschitz解的一些非唯一性和不稳定性结果。对于抛物线系统,此条件(OC)被证明与强多凸性兼容。结果,我们证明了某些\(2 \ times 2 \)的梯度流的初始边界值问题强多凸泛函具有精确的Lipschitz解的弱收敛序列,其弱极限不是弱解。这样的不稳定性结果不能从相应的椭圆系统获得。

更新日期:2020-07-13
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