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 Mat. Sb. (N.S.), 1983, Volume 120(162), Number 3, Pages 396–425 (Mi msb2138)

Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation

S. N. Kruzhkov, A. V. Faminskii

Abstract: In this paper the Cauchy problem for the Korteweg–de Vries equation $u_t+u_{xxx}=uu_x$, $x\in\mathbf R^1$, $0<t<T$, with initial condition $u(0,x)=u_0(x)$ is considered in nonlocal formulation. In the case of an arbitrary initial function $u_0(x)\in L^2(\mathbf R^1)$ the existence of a generalized $L^2$-solution is proved, and its smoothness is studied for $t>0$. A class of well-posed solutions is distinguished among the generalized solutions under consideration, and within this class theorems concerning existence, uniqueness and continuous dependence of solutions on initial conditions are proved.
Bibliography: 28 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 48:2, 391–421

Bibliographic databases:

UDC: 517.946
MSC: 35Q20, 35D05

Citation: S. N. Kruzhkov, A. V. Faminskii, “Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation”, Mat. Sb. (N.S.), 120(162):3 (1983), 396–425; Math. USSR-Sb., 48:2 (1984), 391–421

Citation in format AMSBIB
\Bibitem{KruFam83} \by S.~N.~Kruzhkov, A.~V.~Faminskii \paper Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation \jour Mat. Sb. (N.S.) \yr 1983 \vol 120(162) \issue 3 \pages 396--425 \mathnet{http://mi.mathnet.ru/msb2138} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=691986} \zmath{https://zbmath.org/?q=an:0549.35104|0537.35068} \transl \jour Math. USSR-Sb. \yr 1984 \vol 48 \issue 2 \pages 391--421 \crossref{https://doi.org/10.1070/SM1984v048n02ABEH002682} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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