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

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 $B_{\infty ,q}^0({\mathbb{R}}^d),1\le q\le \infty$ is discontinuous at origin. That is, we can construct a sequence initial data $\{u_0^N\}$ satisfying $||u_0^N|{|}_{B_{\infty ,q}^0({\mathbb{R}}^d)}\to 0,N\to \infty$ such that the corresponding solution $\{u^N\}$ with $u^N(0)=u_0^N$ 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 $N$.

15 pages