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

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.
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
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
\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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    This publication is cited in the following articles:
    1. Kruzhkov S., Faminskii A., “Continuity Properties of Solutions of Some Classes of Time-Dependent Equations”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1983, no. 3, 29–36  mathscinet  zmath  isi
    2. Amy Cohen, Thomas Kappeler, “Solutions to the Korteweg–de Vries Equation with Initial Profile in $L_1^1 (\mathbb{R}) \cap L_N^1 (\mathbb{R}^ + )$”, SIAM J Math Anal, 18:4 (1987), 991  crossref  mathscinet  zmath  isi
    3. Yoshio Tsutsumi, “The Cauchy Problem for the Korteweg–De Vries Equation with Measures as Initial Data”, SIAM J Math Anal, 20:3 (1989), 582  crossref  mathscinet  zmath  isi
    4. J. Ginibre, Y. Tsutsumi, “Uniqueness of Solutions for the Generalized Korteweg–de Vries Equation”, SIAM J Math Anal, 20:6 (1989), 1388  crossref  mathscinet  zmath  isi
    5. A. V. Faminskii, “The Cauchy problem for odd-order quasilinear equations”, Math. USSR-Sb., 68:1 (1991), 31–59  mathnet  crossref  mathscinet  zmath  isi
    6. J. Ginibre, Y. Tsutsumi, G. Velo, “Existence and uniqueness of solutions for the generalized Korteweg de Vries equation”, Math Z, 203:1 (1990), 9  crossref  mathscinet  zmath  isi
    7. Walter Craig, Jonathan Goodman, “Linear dispersive equations of Airy type”, Journal of Differential Equations, 87:1 (1990), 38  crossref  mathscinet  zmath
    8. Gustavo Ponce, Luis Vega, “Nonlinear small data scattering for the generalized Korteweg-de Vries equation”, Journal of Functional Analysis, 90:2 (1990), 445  crossref  mathscinet  zmath
    9. Zhidkov P., “Cauchy-Problem for the Ordinary Korteweg-Devries Differential-Equation with Periodic Initial Data”, Differ. Equ., 26:5 (1990), 591–596  mathnet  mathscinet  zmath  isi
    10. J Ginibre, G Velo, “Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation”, Journal of Differential Equations, 93:1 (1991), 150  crossref  mathscinet  zmath
    11. J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations”, GAFA Geom funct anal, 3:3 (1993), 209  crossref  mathscinet  zmath
    12. J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations”, GAFA Geom funct anal, 3:2 (1993), 107  crossref  mathscinet  zmath
    13. Carlos E. Kenig, Gustavo Ponce, Luis Vega, “Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle”, Comm Pure Appl Math, 46:4 (1993), 527  crossref  mathscinet  zmath
    14. Walter Craig, Thomas Kappeler, Walter Strauss, “Microlocal dispersive smoothing for the Schrödinger equation”, Comm Pure Appl Math, 48:8 (1995), 769  crossref  mathscinet  zmath
    15. Bing-Yu Zhang, “Analyticity of Solutions of the Generalized Korteweg–de Vries Equation with Respect to Their Initial Values”, SIAM J Math Anal, 26:6 (1995), 1488  crossref  mathscinet  zmath  isi
    16. Naumkin P., Shishmarev I., “On Asymptotic-Behavior of Solutions of Generalized Korteweg de Vries Equation as T-]Infinity”, Dokl. Akad. Nauk, 344:2 (1995), 165–167  mathnet  mathscinet  zmath  isi
    17. Faminskii A., “The Cauchy Problem for the Zakharov-Kuznetsov Equation”, Differ. Equ., 31:6 (1995), 1002–1012  mathnet  mathscinet  isi
    18. P. I. Naumkin, I. A. Shishmarev, “Asymptotic behaviour as $t\to \infty$ of the solutions of the generalized Korteweg–de Vries equation”, Sb. Math., 187:5 (1996), 693–733  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. P. E. Zhidkov, “Invariant measures generated by higher conservation laws for the Korteweg–de Vries equations”, Sb. Math., 187:6 (1996), 803–822  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. Zhou Xin, “Strong regularizing effect of integrable systems”, Communications in Partial Differential Equations, 22:3-4 (1997), 503  crossref  mathscinet  zmath
    21. Hongsheng Cai, “Dispersive Smoothing Effects for KdV Type Equations”, Journal of Differential Equations, 136:2 (1997), 191  crossref  mathscinet  zmath
    22. N. S. Bakhvalov, M. I. Zelikin, A. S. Kalashnikov, V. L. Kamynin, O. A. Oleinik, E. Yu. Panov, N. S. Petrosyan, V. M. Tikhomirov, A. V. Faminskii, V. N. Chubarikov, “Stanislav Nikolaevich Kruzhkov (obituary)”, Russian Math. Surveys, 53:5 (1998), 1071–1078  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. Hayashi, N, “Large time asymptotics of solutions to the generalized Korteweg-de Vries equation”, Journal of Functional Analysis, 159:1 (1998), 110  crossref  mathscinet  zmath  isi
    24. A. V. Faminskii, “Mixed problems for the Korteweg–de Vries equation”, Sb. Math., 190:6 (1999), 903–935  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. Hayashi, N, “Large time behavior of solutions for the modified Korteweg-de Vries equation”, International Mathematics Research Notices, 1999, no. 8, 395  crossref  mathscinet  zmath  isi  elib
    26. German Fonsecal, Felipe Linares, Gustavo Ponce, “Global well-posedness for the modified korteweg-de vries equation”, Communications in Partial Differential Equations, 24:3-4 (1999), 683  crossref  mathscinet  zmath
    27. Jacques Printems, “The Stochastic Korteweg–de Vries Equation inL2()”, Journal of Differential Equations, 153:2 (1999), 338  crossref  mathscinet  zmath
    28. Hayashi N., Naumkin P., “On the modified Korteweg de Vries equation”, International Seminar Day on Diffraction, Proceedings, 1999, 146–156  crossref  mathscinet  isi
    29. Zhidkov, P, “Korteweg-de Vries and nonlinear Schroginger equations: Qualitative theory”, Korteweg-de Vries and Nonlinear Schroginger Equations: Qualitative Theory, 1756 (2001), 1  crossref  mathscinet  isi
    30. Hayashi N., Naumkin P., “On the Modified Korteweg-de Vries Equation”, Math. Phys. Anal. Geom., 4:3 (2001), 197–227  crossref  mathscinet  zmath  isi
    31. Maslov V., “Mathematical Aspects of Integral Optics”, Russ. J. Math. Phys., 8:2 (2001), 180–238  mathscinet  zmath  isi
    32. Takaoka H., Tzvetkov N., “On the Local Regularity of the Kadomtsev-Petviashvili-II Equation”, Int. Math. Res. Notices, 2001, no. 2, 77–114  crossref  mathscinet  zmath  isi
    33. Maslov V., “Mathematical Aspects of Integral Optics”, Russ. J. Math. Phys., 8:1 (2001), 83–105  crossref  mathscinet  zmath  isi
    34. Faminskii, AV, “Initial boundary-value problem in a half-strip for the Korteweg-de Vries equation in fractional-order Sobolev spaces”, Communications in Partial Differential Equations, 29:11–12 (2004), 1653  crossref  mathscinet  zmath  isi  elib
    35. C.E.. Kenig, Gustavo Ponce, Luis Vega, “The Cauchy problem for quasi-linear Schr�dinger equations”, Invent. math, 158:2 (2004), 343  crossref  mathscinet  zmath
    36. Ademir Fernando Pazoto, “Unique continuation and decay for the Korteweg-de Vries equation with localized damping”, ESAIM: COCV, 11:3 (2005), 473  crossref  mathscinet  zmath
    37. Kappeler T., Perry P., Shubin M., Topalov P., “The Miura Map on the Line”, Int. Math. Res. Notices, 2005, no. 50, 3091–3133  crossref  mathscinet  zmath  isi  elib
    38. Liana L. Dawson, “Uniqueness properties of higher order dispersive equations”, Journal of Differential Equations, 236:1 (2007), 199  crossref  mathscinet  zmath
    39. L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega, “On uniqueness properties of solutions of the k-generalized KdV equations”, Journal of Functional Analysis, 244:2 (2007), 504  crossref  mathscinet  zmath
    40. Chengchun Hao, Ling Hsiao, Baoxiang Wang, “Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces”, Journal of Mathematical Analysis and Applications, 328:1 (2007), 58  crossref  mathscinet  zmath
    41. Bona, JL, “Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane”, Annales de l Institut Henri Poincare-Analyse Non Lineaire, 25:6 (2008), 1145  crossref  mathscinet  zmath  isi  elib
    42. J LEVANDOSKY, M SEPULVEDA, O VERAVILLAGRAN, “Gain of regularity for the KP-I equation☆”, Journal of Differential Equations, 245:3 (2008), 762  crossref  mathscinet  zmath
    43. J. A. Barceló, J.M. Bennett, A. Ruiz, M. C. Vilela, “Local smoothing for Kato potentials in three dimensions”, Math Nachr, 282:10 (2009), 1391  crossref  mathscinet  zmath  isi
    44. Margareth Alves, Mauricio Sepúlveda, Octavio Vera, “Smoothing properties for the higher-order nonlinear Schrödinger equation with constant coefficients”, Nonlinear Analysis: Theory, Methods & Applications, 71:3-4 (2009), 948  crossref  mathscinet  zmath
    45. F. Linares, A.F. Pazoto, “Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane”, Journal of Differential Equations, 246:4 (2009), 1342  crossref  mathscinet  zmath
    46. Fokas A.S., “Lax pairs: a Novel Type of Separability”, Inverse Probl., 25:12 (2009), 123007  crossref  mathscinet  zmath  adsnasa  isi
    47. S. I. Pokhozhaev, “On the Singular Solutions of the Korteweg–de Vries Equation”, Math. Notes, 88:5 (2010), 741–747  mathnet  crossref  crossref  mathscinet  isi
    48. Pohozaev S.I., “On a Class of Singular Solutions to the Korteweg-de Vries Equation”, Dokl. Math., 82:3 (2010), 936–938  crossref  mathscinet  zmath  isi  elib
    49. Nikolai A. Larkin, Eduardo Tronco, “Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip”, Journal of Differential Equations, 2012  crossref  mathscinet
    50. Audiard C., “Dispersive Smoothing for the Euler-Korteweg Model”, SIAM J. Math. Anal., 44:4 (2012), 3018–3040  crossref  mathscinet  zmath  isi
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    52. Akhunov T., “Local Well-Posedness of Quasi-Linear Systems Generalizing KdV”, Commun. Pure Appl. Anal, 12:2 (2013), 899–921  crossref  mathscinet  zmath  isi
    53. N.A.. Larkin, “Exponential decay of the -norm for the 2D Zakharov-Kuznetsov equation on a half-strip”, Journal of Mathematical Analysis and Applications, 2013  crossref  mathscinet
    54. Pedro Isaza, Felipe Linares, Gustavo Ponce, “On Decay Properties of Solutions of the k-Generalized KdV Equation”, Commun. Math. Phys, 2013  crossref  mathscinet
    55. Bing-Yu Zhang, Xiangqing Zhao, “Boundary smoothing properties of the Kawahara equation posed on the finite domain”, Journal of Mathematical Analysis and Applications, 2014  crossref  mathscinet
    56. Pedro Isaza, Felipe Linares, Gustavo Ponce, “On the Propagation of Regularity and Decay of Solutions to thek-Generalized Korteweg-de Vries Equation”, Communications in Partial Differential Equations, 2014, 1412030655  crossref  mathscinet
    57. A. P. Antonova, A. V. Faminskii, “On the Regularity of Solutions of the Cauchy Problem for the Zakharov–Kuznetsov Equation in Hölder Norms”, Math. Notes, 97:1 (2015), 12–20  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    58. G. G. Doronin, N. A. Larkin, “Stabilization of Regular Solutions for the Zakharov–Kuznetsov Equation posed on Bounded Rectangles and on a Strip”, Proceedings of the Edinburgh Mathematical Society, 2015, 1  crossref  mathscinet  zmath
    59. A. R. Khashimov, “Vtoraya kraevaya zadacha dlya nestatsionarnogo uravneniya tretego poryadka sostavnogo tipa”, Matematicheskie zametki SVFU, 24:4 (2017), 76–86  mathnet  crossref  elib
    60. A. V. Faminskii, “O vnutrennei regulyarnosti reshenii dvumernogo uravneniya Zakharova–Kuznetsova”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 65, no. 3, Rossiiskii universitet druzhby narodov, M., 2019, 513–546  mathnet  crossref
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