Everywhere regularity of certain nonlinear diffusion systems |
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Authors: | Maxim Trokhimtchouk |
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Institution: | 1. University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA, 94720-3840, USA
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Abstract: | In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where ${\Phi(z)}$ is a strictly convex function. I show that when ${\Phi}$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems. |
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