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This article is cited in 59 scientific papers (total in 60 papers)
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.
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Mathematics of the USSR-Sbornik, 1984, 48:2, 391–421
Bibliographic databases:
 
UDC:
517.946
MSC: 35Q20, 35D05
Received: 27.05.1982
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
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\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
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=691986}
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\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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