Geodesic
Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function
Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 119-128.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Korteweg–de Vries equation, Cauchy problem, Burgers equation, Banach space, gas-dynamic problem, line of weak discontinuity, Bochner measurable mapping. 
@article{MZM_2008_83_1_a12,
     author = {A. V. Faminskii},
     title = {Cauchy {Problem} for the {Korteweg--de~Vries} {Equation} in the {Case} of a {Nonsmooth} {Unbounded} {Initial} {Function}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {119--128},
     publisher = {mathdoc},
     volume = {83},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a12/}
}
TY  - JOUR
AU  - A. V. Faminskii
TI  - Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function
JO  - Matematičeskie zametki
PY  - 2008
SP  - 119
EP  - 128
VL  - 83
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a12/
LA  - ru
ID  - MZM_2008_83_1_a12
ER  - 
%0 Journal Article
%A A. V. Faminskii
%T Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function
%J Matematičeskie zametki
%D 2008
%P 119-128
%V 83
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a12/
%G ru
%F MZM_2008_83_1_a12
A. V. Faminskii. Cauchy Problem for the Korteweg--de~Vries Equation in the Case of a Nonsmooth Unbounded Initial Function. Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 119-128. http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a12/

[1] A. M. Ilin, D. I. Neudachin, “O vliyanii maloi dissipatsii na resheniya giperbolicheskoi sistemy, imeyuschei slabyi razryv”, Dokl. RAN, 351:5 (1996), 583–585 | MR | Zbl

[2] D. I. Neudachin, “O vliyanii maloi dissipatsii na reshenie zadachi o bezudarnom szhatii ploskogo sloya gaza”, ZhVM i MF, 39:2 (1999), 332–340 | MR | Zbl

[3] S. V. Zakharov, A. M. Ilin, “Ot slabogo razryva k gradientnoi katastrofe”, Matem. sb., 192:10 (2001), 3–18 | MR | Zbl

[4] S. V. Zakharov, “Zarozhdenie udarnoi volny v odnoi zadache Koshi dlya uravneniya Byurgersa”, ZhVM i MF, 44:3 (2004), 536–542 | MR | Zbl

[5] A. Menikoff, “The existence of unbounded solutions of the Korteweg–de Vries equation”, Comm. Pure Appl. Math., 25:4 (1972), 407–432 | DOI | MR | Zbl

[6] I. N. Bondareva, “Uravnenie Kortevega–de Friza v klassakh rastuschikh funktsii s zadannoi asimptotikoi pri $|x|\to\infty$”, Matem. sb., 122:2 (1983), 131–141 | MR | Zbl

[7] I. N. Bondareva, M. A. Shubin, “O edinstvennosti resheniya zadachi Koshi dlya uravneniya Kortevega–de Friza v klassakh rastuschikh funktsii”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1985, no. 3, 35–38 | MR | Zbl

[8] I. N. Bondareva, M. A. Shubin, “Uravneniya tipa Kortevega–de Friza v klassakh rastuschikh funktsii”, Tr. sem. im. I. G. Petrovskogo, 14 (1989), 45–56 | MR | Zbl

[9] C. E. Kenig, G. Ponce, L. Vega, “Global solutions for the KdV equation with unbounded data”, J. Differential Equations, 139:2 (1997), 339–364 | DOI | MR | Zbl

[10] Kh. Gaevskii, K. Groger, K. Zakharias, Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR | Zbl

[11] A. V. Faminskii, “Zadacha Koshi dlya uravneniya Kortevega–de Friza i ego obobschenii”, Tr. sem. im. I. G. Petrovskogo, 13 (1988), 56–105 | MR | Zbl

[12] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl