Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function

A. V. Faminskii

Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 119-128

Résumé

In the strip $\Pi=(-1,0)\times\mathbb R$, we establish the existence of solutions of the Cauchy problem for the Korteweg–de Vries equation $u_t+u_{xxx}+uu_x=0$ with initial condition either 1) $u(-1,x)=-x\theta(x)$, or 2) $u(-1,x)=-x\theta(-x)$, where $\theta$ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for $t\in(-1,0)$ and rapidly decreasing as $x\to+\infty$. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.

Détail

Keywords: Korteweg–de Vries equation, Cauchy problem, Burgers equation, Banach space, gas-dynamic problem, line of weak discontinuity, Bochner measurable mapping.