Weak solutions to a hyperbolic-elliptic problem

초록

We prove the existence of infinitely many local-in-time weak solutions to the initial-boundary value problem for a class of hyperbolic-elliptic equations in dimension n >= 2 when the range of the magnitude of the initial spatial gradient overlaps with the unstable elliptic regime. Such solutions are extracted from the method of convex integration in a Baire category setup; they are smooth outside the phase mixing zone that is determined by a modified hyperbolic evolution, continuous on the space-time domain, and Lipschitz continuous in terms of the spatial variables. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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