On the optimal Sobolev threshold for evolution equations with rough nonlinearities

Consider a general evolution equation of the form

 

$\partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u})$

 

where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type, and the nonlinear term $F$ has limited regularity (e.g. it is Hölder continuous up to a certain order). In this talk, I will describe a robust set of techniques which can be used in many cases to predict the highest possible Sobolev exponent $s=s(q,d)$ for which the above evolution can be well-posed in $W_x^{s,q}(\mathbb{R}^d)$. I will discuss how these principles can be rigorously implemented in the model cases of the nonlinear Schr"odinger and nonlinear heat equations. More precisely, we are able to show that the nonlinear heat equation

 

$\partial_tu-\Delta u=|u|^{p-1}u$

 

is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\}<s<2+p+\frac{1}{q}$ and is strongly ill-posed when $s\geq 2+p+\frac{1}{q}$ and $p-1\not\in 2\mathbb{N}$  in the sense of non-existence of solutions even for smooth, small and compactly supported data. When $q=2$, we establish the same ill-posedness result for the nonlinear Schrödinger equation and the corresponding well-posedness result when $p\geq \frac{3}{2}$. Identifying the optimal Sobolev threshold for even a single non-algebraic $p>1$ has been a longstanding folklore open problem in the literature. As an amusing corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking $d$ sufficiently large relative to $p$ that there are nonlinear Schrödinger equations which are ill-posed in every Sobolev space $H_x^s(\mathbb{R}^d)$. This is based on a joint work with Mitchell Taylor.