Abstract:
In this paper the Cauchy problem for the Korteweg–de Vries equation ut+uxxx=uux, x∈R1, 0<t<T, with initial condition u(0,x)=u0(x) is considered in nonlocal formulation. In the case of an arbitrary initial function u0(x)∈L2(R1) the existence of a generalized L2-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.
Citation:
S. N. Kruzhkov, A. V. Faminskii, “Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation”, Math. USSR-Sb., 48:2 (1984), 391–421
\Bibitem{KruFam83}
\by S.~N.~Kruzhkov, A.~V.~Faminskii
\paper Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation
\jour Math. USSR-Sb.
\yr 1984
\vol 48
\issue 2
\pages 391--421
\mathnet{http://mi.mathnet.ru/eng/sm2138}
\crossref{https://doi.org/10.1070/SM1984v048n02ABEH002682}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=691986}
\zmath{https://zbmath.org/?q=an:0549.35104|0537.35068}
Linking options:
https://www.mathnet.ru/eng/sm2138
https://doi.org/10.1070/SM1984v048n02ABEH002682
https://www.mathnet.ru/eng/sm/v162/i3/p396
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