I'm studying the viscosity solutions to the Hamilton-Jacobi equations by Evans's book. If we apply the method of vanishing viscosity and add the regularizing term, we get the following quasilinear parabolic equation: $$ u_t^{\varepsilon} + H(D_x u^\varepsilon, x) = \varepsilon \Delta u^\varepsilon \\ u^\varepsilon(x,0) = g$$ Evans simply puts it that these turn out to have smooth solutions. I've been wondering why that is true. I tried to find something in the book by Ladyzhenskaya "Linear and Quasilinear Equations of Parabolic Type", but seems like there the theory is much more general than I need, so it would take me a lot of time to pass through. Could anybody give me some reference or explanation of this fact? In this case $H$ is a smooth function, and $g$ is continuous.
2026-03-25 12:55:14.1774443314
Reference to Quasilinear Parabolic PDE
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See Lemma 4.3 and the surrounding discussion in [1]. There, Evans uses a result of Friedman's in [2]. The paper [2] can be downloaded here.
[1] Evans, Lawrence C., On solving certain nonlinear partial differential equations by accretive operator methods, Isr. J. Math. 36, 225-247 (1980). ZBL0454.35038.
[2] Friedman, Avner, The Cauchy problem for first order partial differential equations, Indiana Univ. Math. J. 23, 27-40 (1973). ZBL0243.35014.