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On evolutionary problems with a-priori bounded gradients

Banff International Research Station for Mathematical Innovation and Discovery

We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely $L^1$-coercivity. We use the concept of renormalized solutions and higher differentiability techniques to prove existence and uniqueness of weak solution with $L^1$-integrable flux for all values of a positive model parameter $a$. If this parameter is smaller than $2/(d+1)$, where $d$ denotes the spatial dimension, we obtain higher integrability of the flux. We also relate the studied problem to problems in fluid mechanics.

Mathematics; Partial differential equations; Fluid mechanics; Fluid dynamics

video/mp4

Author affiliation: Charles University, Faculty of Mathematics and Physics

2021-05-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/78419

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/