Ill-posedness for the Hamilton-Jacobi equation in Besov spaces $B^0_{\infty,q}$

TL;DR

This paper demonstrates the ill-posedness of the Hamilton-Jacobi equation in certain Besov spaces by showing the discontinuity of the solution map at zero initial data.

Contribution

It establishes the discontinuity of the solution map for the Hamilton-Jacobi equation in Besov spaces $B^0_{ abla,q}$, revealing ill-posedness in these function spaces.

Findings

Solution map discontinuous at zero initial data

Constructed initial data sequences with vanishing norm

Solutions remain bounded away from zero in norm

Abstract

In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \bbal\bca \pa_tu-\De u=|\na u|^2,\quad t>0, \ x\in \R^d,\\ u(0,x)=u_0, \quad \quad x\in \R^d. \eca\end{align*} We show that the solution map in Besov spaces $B^0_{\infty,q}(\R^d),1\leq q\leq \infty$ is discontinuous at origin. That is, we can construct a sequence initial data $\{u^N_0\}$ satisfying $||u^N_0||_{B^0_{\infty,q}(\R^d)}\rightarrow 0, \ N\rightarrow \infty$ such that the corresponding solution $\{u^N\}$ with $u^N(0)=u^N_0$ satisfies \bbal ||u^N||_{L^\infty_T(B^0_{\infty,q}(\R^d))}\geq c_0, \qquad \forall \ T>0, \quad N\gg 1, \end{align*} with a constant $c_0>0$ independent of $N$.