# KdV/KP-II equation with upper semicontinuous initial data and viscosity solutions

In the article "KP governs random growth off a 1-dimensional substrate", they study the KP-II equation: the function $$\phi(t,x,r)=\partial_{r}^{2}\log(F)$$ satisfies

$$\partial_{t}\phi+\frac{1}{2}\partial_{r}(\phi)^{2}+\frac{1}{12}\partial_{r}^{3}(\phi)+\frac{1}{4}\partial_{r}^{-1}\partial_{x}^{2}\phi=0$$

or in the Hirota-form

$$F\partial_{tr}^{2}F-\partial_{t}F\partial_{r}F+\frac{1}{12}F\partial_{r}^{4}F-\frac{1}{3}\partial_{r}F\partial_{r}^{3}F+\frac{1}{4}(\partial_{r}^{2}F)^{2}+\frac{1}{4}F\partial_{x}^{2}F-\frac{1}{4}(\partial_{x}F)^{2}=0$$

with the following initial data (see example 1.6): first consider the space of upper semicontinuous functions

$$UC=\{h:\mathbb{R}\to [-\infty,\infty): \text{ h is upper semicontinuous}, h(x)\leq A+B|x|, A,B>0, h\not\equiv -\infty\},$$ we then fix some $$h_{0}(x)\in UC$$ and set

$$\phi(0,x,r):=0,\text{ for }r\geq h_{0}(x),~~~~~,\phi(0,x,r):=-\infty,\text{ for }r< h_{0}(x).$$

Q1: Has anything close to this type of initial data been covered in the literature? If not, do you have any suggestions on where to start e.g. some article/method to try to modify for this particular initial data? How would you study this problem?

In https://www.math.ucla.edu/~tao/Dispersive/misc.html#KP-II, they list various results for initial data. The weakest regularity seems to be negative Sobolev space $$H^{s}(R)$$.

For example, in "The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below L2" they prove locally well-posedness for $$s>-\frac{1}{90}$$.

Attempts 1)One context where such initial data shows up is in Hamilton-Jacobi equations. For example, in

I want to try the method described in (a) where they take initial conditions $$g_{n}\in C$$ converging to the UC function $$g$$ and consider the corresponding viscosity solutions. In "Asymptotic behavior of the generalized Korteweg–de Vries–Burgers equation" is one place where I found the use of viscosity solutions being used but the IC is in $$H^2$$

Q2: Do you think the viscosity solutions can be helpful here?

2)I am also trying to see if the notion of differential inclusions can be helpful here. To at least a provide a notion of well-posedness. In "Dirichlet Problems for some Hamilton-Jacobi Equations with Inequality Constraints", they study similar initial data. There they work with another notion of viscosity solutions: the unique generalized solution in the Barron-Jensen/Frankowska sense (a weaker concept of viscosity solution introduced by Crandall, Evans, and Lions in for continuous solutions adapted to the case when solution is only semicontinuous).

Q3: Has the notion of differential inclusions applied to KdV? If not, do you see any obstacles?