Everywhere regularity of certain nonlinear diffusion systems

Trokhimtchouk, Maxim
March 2010
Calculus of Variations & Partial Differential Equations;Mar2010, Vol. 37 Issue 3/4, p407
Academic Journal
In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form 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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