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.


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.

  • $\begingroup$ Thank you! This is exactly what I was looking for. These theorems basically provide another way to ensure the existence of the viscosity solution, defined by this vanishing viscosity method, which basically differ from the dynamic programming method, that Evans uses later in the chapter of PDE. $\endgroup$ – Proton May 4 '17 at 17:01
  • $\begingroup$ @Proton: no worries. In general, if you obtain an answer to your question, consider accepting it (though it is entirely reasonable to wait some time before doing so). $\endgroup$ – parsiad May 4 '17 at 19:43

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