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Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows

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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 \(2\times 2\) 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.

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Acknowledgements

I would like to thank the anonymous referee for helpful suggestions that improve the presentation of the paper.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Baisheng Yan

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  1. Baisheng Yan
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Correspondence to Baisheng Yan.

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Communicated by J. Ball.

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Yan, B. Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows. Calc. Var. 59, 123 (2020). https://doi.org/10.1007/s00526-020-01785-7

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