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This is a soft question. For my background, I am an undergraduate math major just finishing my first course in PDE. In my PDE class, we have been learning about viscosity solutions.

I am following the definitions in these notes

We say a continuous function $u : [0, T ] \times \Omega → \mathbb{R}$ is a subsolution ( alt supersolution) of the equation $u_t + H(x,t,u,\nabla u) =0$, if every time there exists a $C^1$ function $\varphi$ for which there is a point $(t_0, x_0) \in (0, T ] \times \Omega$ and $r>0$, such that $\varphi (t_0, x_0) = u(t_0, x_0)$, $\varphi (t,x)≥u(t,x)$ (alt $≤$) for all $(t,x) \in(t_0 −r,t_0]×B_r(x_0)$, then $\varphi_t +H(t,x,\varphi,\nabla \varphi)≤0$ (alt $≥0$).

A continuous function $u$ is a viscosity solution in $(0, T ] × Ω$ when it is both a subsolution and a supersolution.

Evans also discusses these solutions in Chapter 10 of his book on PDE. Evans motivates viscosity solutions by looking at a sequence of functions $u^\varepsilon$ that are solutions to $u_t + H(x,t,u,\nabla u) - \varepsilon \Delta u =0$. This equation has smooth solutions, because the Laplacian regularizes the Hamilton-Jacobi equation. Often we can assure that the sequence $u^\varepsilon$ is bounded and equicontinuous, giving some locally uniformly convergent subsequence, and these turn out to be viscosity solutions. (Though not all viscosity solutions arise in this way)

I am also aware that viscosity solutions can be defined for somewhat more general non-linear PDEs.

My questions are the following:

  • Why are such generalized solutions interesting? Are they only interesting because we can prove the existence and uniqueness of such solutions? Or do viscosity solutions to Hamilton-Jacobi equations actually model anything?

  • Are these solutions ever useful in applications? This may be just a more specific form of the above question.

  • Why should the notion of viscosity solution for Hamilton-Jacobi equations be the "right" definition of generalized solution?

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    $\begingroup$ I can see I've been down voted twice now. If anyone has suggestions on how I could improve the question or what's wrong with it, please let me know. I am quite interested in its answer. $\endgroup$ – msm Jun 7 '17 at 14:27
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The method of vanishing viscosity is not a particularly good way to think about viscosity solutions. The reason viscosity solutions are so useful is that they satisfy a maximum principle (or rather a comparison principle), which gives them very nice uniqueness, existence, and stability results (making it easy to pass to limits, for instance). For these reasons, viscosity solutions are almost always the correct notion of solution in applications, which include things like optimal control theory and geometric flows like mean curvature motion, to name a few.

Viscosity solutions apply to fully nonlinear equations for which no other notion of solution is available (e.g., classical solutions do not exist, and weak distributional solutions do not make sense). Without viscosity solutions, we would have no way to rigorously make sense of a very broad class of PDEs. I could go on, but others have written more elegant answers to this question elsewhere:

https://mathoverflow.net/questions/59449/why-are-viscosity-solutions-useful-solutions

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