Geodesic
Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation
Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 391-421 
S. N. Kruzhkov; A. V. Faminskii. Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 391-421. http://geodesic.mathdoc.fr/item/SM_1984_48_2_a7/
@article{SM_1984_48_2_a7,
     author = {S. N. Kruzhkov and A. V. Faminskii},
     title = {Generalized solutions of the {Cauchy} problem for the {Korteweg-de} {Vries} equation},
     journal = {Sbornik. Mathematics},
     pages = {391--421},
     year = {1984},
     volume = {48},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1984_48_2_a7/}
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In this paper the Cauchy problem for the Korteweg–de Vries equation $u_t+u_{xxx}=uu_x$, $x\in\mathbf R^1$, $0, 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.