Lipschitz Solutions for the Gradient Flow of Polyconvex Functionals

2019/01/17 by Baisheng Yan, Yan, Baisheng
Mathematics · #35D30 (Primary) #35F50 #35K40 #35K51 #49A20 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1901.05989

openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this sequel to a previous paper, we construct certain smooth strongly polyconvex functions F on \mathbb M^{2× 2} such that σ=DF satisfies the Condition (OC) in that paper. As a result, we show that the initial-boundary value problem for the gradient flow of such polyconvex energy functionals is highly ill-posed even for some smooth initial-boundary data in the sense that the problem possesses a weakly* convergent sequence of Lipschitz weak solutions whose limit is not a weak solution.

Citations

• Convexity conditions and existence theorems in nonlinear elasticity

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