Mixed problems for the Korteweg–de Vries equation

A. V. Faminskii

Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 903-935

Résumé

Results are established concerning the non-local solubility and wellposedness in various function spaces of the mixed problem for the Korteweg–de Vries equation $$ u_t+u_{xxx}+au_x+uu_x=f(t,x) $$ in the half-strip $(0,T)\times(-\infty,0)$. Some a priori estimates of the solutions are obtained using a special solution $J(t,x)$ of the linearized KdV equation of boundary potential type. Properties of $J$ are studied which differ essentially as $x\to+\infty$ or $x\to-\infty$. Application of this boundary potential enables us in particular to prove the existence of generalized solutions with non-regular boundary values.

Bibliographie

[1] Bui An Ton, “Initial boundary-value problems for the Korteweg–de Vries equation”, J. Differential Equations, 25:3 (1977), 288–310 | DOI | MR

[2] Khablov V. V., O nekotorykh korrektnykh postanovkakh granichnykh zadach dlya uravneniya Kortevega–de Friza, Preprint In-ta matem. SO AN SSSR, Novosibirsk, 1979

[3] Kruzhkov S. N., Faminskii A. V., “Obobschennye resheniya uravneniya Kortevega–de Friza”, Dokl. AN SSSR, 261:6 (1981), 1296–1298 | MR | Zbl

[4] Bubnov B. A., “Razreshimost v tselom nelineinykh granichnykh zadach dlya uravneniya Kortevega–de Friza v ogranichennoi oblasti”, Differents. uravneniya, 16:1 (1980), 34–41 | MR | Zbl

[5] Bona J. L., Winther R., “The Korteweg–de Vries equation, posed in a quarter-plane”, SIAM J. Math. Anal., 14:6 (1983), 1056–1106 | DOI | MR | Zbl

[6] Faminskii A. V., “Smeshannaya zadacha v polupolose dlya uravneniya Kortevega–de Friza i ego obobschenii”, Tr. MMO, 51, URSS, M., 1988, 54–94 | MR

[7] Bona J. L., Winther R., “The Korteweg–de Vries equation in a quarter-plane, continuous dependence results”, Differential Integral Equations, 2:2 (1989), 228–250 | MR | Zbl

[8] Faminskii A. V., “Nonlocal well-posedness of a mixed problem in a half-strip for the Korteweg–de Vries equation”, Vestn. RUDN. Ser. matem., 3:1 (1996), 156–176

[9] Faminskii A. V., “Nelokalnaya korrektnost kraevykh zadach dlya uravneniya Kortevega–de Friza i ego obobschenii”, Materialy Mezhdunarodnoi konferentsii i Chebyshevskikh chtenii, posvyaschennykh 175-letiyu so dnya rozhdeniya P. L. Chebysheva, T. 2, Izd-vo MGU, M., 1996, 341–344 | MR

[10] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR

[11] Cattabriga L., “Un problema al contorno per una equazione parabolica di ordine dispari”, Ann. Scuola Norm. Sup. Pisa Sci. Fiz. Mat., 13:2 (1959), 163–203 | MR | Zbl

[12] Dzhuraev T. D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipov, Fan, Tashkent, 1979 | MR | Zbl

[13] Solonnikov V. A., “Apriornye otsenki dlya uravnenii vtorogo poryadka parabolicheskogo tipa”, Tr. MIAN, 70, Nauka, M., 1964, 133–212 | MR | Zbl

[14] Kenig C. E., Ponce G., Vega L., “Well-posedness of the initial value problem for the Korteweg–de Vries equation”, J. Amer. Math. Soc., 4:2 (1991), 323–347 | DOI | MR | Zbl

[15] Kenig C. E., Ponce G., Vega L., “Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle”, Comm. Pure Appl. Math., 43 (1993), 527–620 | DOI | MR

[16] Kruzhkov S. N., Faminskii A. V., “Obobschennye resheniya zadachi Koshi dlya uravneniya Kortevega–de Friza”, Matem. sb., 120:3 (1983), 396–425 | MR | Zbl

[17] Kato T., “On the Cauchy problem for the (generalized) Korteweg–de Vries equation”, Appl. Math., Suppl. Stud., 8 (1983), 93–128 | MR | Zbl

[18] Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR | Zbl

[19] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[20] Berg I., Lefstrem I., Interpolyatsionnye neravenstva. Vvedenie, Mir, M., 1980

[21] Stein E. M., “Oscillatory integrals in Fourier analysis”, Beijing lectures in harmonic analysis, Princeton Univ. Press, Princeton, 1986, 307–355 | MR

[22] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR