Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity

Esteban, Juan R.; Marcati, Pierangelo

doi:10.1090/S0002-9947-1994-1214784-8

Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity
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by Juan R. Esteban and Pierangelo Marcati

Trans. Amer. Math. Soc. 342 (1994), 501-521

DOI: https://doi.org/10.1090/S0002-9947-1994-1214784-8

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Abstract:

We shall consider a model problem for the fully nonlinear parabolic equation \[ {u_t} + F(x,t,u,Du,\varepsilon {D^2}u) = 0\] and we study both the approximating degenerate second order problem and the related first order equation, obtained by the limit as $\varepsilon \to 0$. The strong convergence of the gradients is provided by semiconcavity unilateral bounds and by the supremum bounds of the gradients. In this way we find solutions in the class of viscosity solutions of Crandall and Lions.

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