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Comptes Rendus. Mathématique
Partial differential equations
A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation
Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 39-45.

Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter.

L’équation de KdV est considérée comme un modèle approximatif pour des ondes longues de faible amplitude à la surface libre d’un fluide non visqueux. On montre qu’il y a une densité de moment approximative associée à l’équation de KdV, et que la différence entre cette densité et la densité de de moment physique dérivée dans le contexte du système d’Euler peut être estimée en fonction du paramètre d’onde longue.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.143
Samer Israwi 1; Henrik Kalisch 2

1 Lebanese University, Faculty of Sciences 1, Department of Mathematics, Hadath-Beirut, Lebanon
2 Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway
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Samer Israwi; Henrik Kalisch. A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 39-45. doi : 10.5802/crmath.143. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.143/

[1] Mark J. Ablowitz; Harvey Segur On the evolution of packets of water waves, J. Fluid Mech., Volume 92 (1979), pp. 691-715 | Article | MR: 544892 | Zbl: 0413.76009

[2] Alfatih Ali; Henrik Kalisch Energy balance for undular bores, C. R. Méc. Acad. Sci. Paris, Volume 338 (2010) no. 2, pp. 67-70 | Zbl: 1300.76010

[3] Alfatih Ali; Henrik Kalisch Mechanical balance laws for Boussinesq models of surface water waves, J. Nonlinear Sci., Volume 22 (2012) no. 3, pp. 371-398 | MR: 2927764 | Zbl: 1253.35113

[4] Alfatih Ali; Henrik Kalisch On the formulation of mass, momentum and energy conservation in the KdV equation, Acta Appl. Math., Volume 133 (2014) no. 1, pp. 113-131 | MR: 3255079 | Zbl: 1310.35206

[5] Charles J. Amick Regularity and uniqueness of solutions to the Boussinesq system of equations, J. Differ. Equations, Volume 54 (1984) no. 2, pp. 231-247 | Article | MR: 757294 | Zbl: 0557.35074

[6] Jerry L. Bona; Min Chen; Jean-Claude Saut Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory, Nonlinearity, Volume 17 (2004) no. 3, pp. 925-952 | Article | MR: 2057134 | Zbl: 1059.35103

[7] Jerry L. Bona; Thierry Colin; David Lannes Long wave approximations for water waves, Arch. Ration. Mech. Anal., Volume 178 (2005) no. 3, pp. 373-410 | Article | MR: 2196497 | Zbl: 1108.76012

[8] Jerry L. Bona; Ronald B. Smith The initial value problem for the Korteweg–de Vries equation, Proc. R. Soc. Lond., Ser. A, Volume 278 (1975) no. 1287, pp. 555-601 | MR: 385355 | Zbl: 0306.35027

[9] Adrian Constantin; David Lannes The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal., Volume 192 (2009) no. 1, pp. 165-186 | Article | MR: 2481064 | Zbl: 1169.76010

[10] Walter Craig An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits, Commun. Partial Differ. Equations, Volume 10 (1985) no. 8, pp. 787-1003 | Article | MR: 795808 | Zbl: 0577.76030

[11] Samer Israwi Variable depth KdV equations and generalizations to more nonlinear regimes, ESAIM, Math. Model. Numer. Anal., Volume 44 (2010) no. 2, pp. 347-370 | Article | Numdam | MR: 2655953 | Zbl: 1258.76040

[12] Samer Israwi; Henrik Kalisch Approximate conservation laws in the KdV equation, Physics Letters A, Volume 383 (2019) no. 9, pp. 854-858 | Article | MR: 3926679

[13] Anna Karczewska; Piotr Rozmej; Eryk Infeld Energy invariant for shallow-water waves and the Korteweg–de Vries equation: Doubts about the invariance of energy, Phys. Rev. E, Volume 92 (2015) no. 5, 053202, 15 pages | Article | MR: 3591119

[14] Anna Karczewska; Piotr Rozmej; Eryk Infeld; George Rowlands Adiabatic invariants of the extended KdV equation, Phys. Lett., Volume 381 (2017) no. 4, pp. 270-275 | Article | MR: 3583641 | Zbl: 1372.35275

[15] David Lannes Well-posedness of the water-waves equations, J. Am. Math. Soc., Volume 18 (2005) no. 3, pp. 605-654 | Article | MR: 2138139 | Zbl: 1069.35056

[16] David Lannes The Water Waves Problem. Mathematical analysis and asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, 2013 | Zbl: 1410.35003

[17] Guido Schneider; C. Eugene Wayne The long-wave limit for the water wave problem. I. The case of zero surface tension, Commun. Pure Appl. Math., Volume 53 (2000) no. 12, pp. 1475-1535 | Article | MR: 1780702 | Zbl: 1034.76011

[18] Maria Elena Schonbek Existence of solutions for the Boussinesq system of equations, J. Differ. Equations, Volume 42 (1981), pp. 325-352 | Article | MR: 639225 | Zbl: 0476.35067

[19] Sijue Wu Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., Volume 130 (1997) no. 1, pp. 39-72 | MR: 1471885 | Zbl: 0892.76009

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