On the optimal Sobolev threshold for evolution equations with rough nonlinearities

TL;DR

The article introduces a robust framework to predict sharp Sobolev thresholds for evolution equations with rough nonlinearities, and confirms it on the nonlinear heat and Schrödinger equations. Central to the approach is a novel nonlinear bound for $|u|^{p-1}u$ in Sobolev spaces, established by approximating $u$ with piecewise-linear functions and leveraging Besov-type tools to control high-regularity maps. The authors derive precise well-posedness results: NLH is locally well-posed in $W^{s,q}$ for $ ext{max}\{0,s_cig ext}$ $<s<p+2+rac{1}{q}$, with ill-posedness at the corresponding endpoint; NLS is locally well-posed in $H^s$ for $ ext{max}\{0,s_cig ext}$ $<s<p+rac{5}{2}$ (with 1D improvements) and ill-posed beyond this threshold. They further provide a thorough time-truncated Schrödinger analysis to obtain a priori bounds, construct solutions via frequency envelope methods, and prove dimension-independent ill-posedness results, implying ill-posedness in every $H^s$ for certain high dimensions. The paper also discusses broader applicability to general dispersion relations, derivative nonlinearities, and wave-type models, suggesting wide potential impact on global dynamics and regularity propagation in rough evolution equations.

Abstract

In this article we are concerned with evolution equations of the form \begin{equation*} \partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u}) \end{equation*} where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type and the nonlinear term $F$ is of limited regularity. Our objective is to develop a robust set of principles which can be used in many cases to predict the \emph{highest} Sobolev exponent $s=s(q,d)$ for which the above evolution is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ (necessarily restricting to $q=2$ for dispersive problems). We will confirm the validity of these principles for two of the most important model problems; namely, the nonlinear Schrödinger and heat equations. More precisely, we will prove that the nonlinear heat equation \begin{equation*} \partial_tu-Δu=\pm |u|^{p-1}u, \hspace{5mm} p>1, \end{equation*} is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\}<s<2+p+\frac{1}{q}$ and is \emph{strongly ill-posed} when $s\geq \max\{s_c,2+p+\frac{1}{q}\}$ 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 $p>1$ was a rather longstanding open problem in the literature. As an immediate corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking $d\gg p$ that there are nonlinear Schrödinger equations which are ill-posed in \emph{every} Sobolev space $H_x^s(\mathbb{R}^d)$.

Theorems & Definitions (83)

• Theorem 1.1: MR1667895MR1691575MR1946761
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Corollary 1.9
• Theorem 1.10