Two-parameter asymptotics in a bisingular Cauchy problem for a parabolic equation

S. V. Zakharov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 94-103

Résumé

The Cauchy problem for a quasilinear parabolic equation with a small parameter $\varepsilon$ at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable $x/\rho$, where $\rho$ is another small parameter. This problem statement is of interest in applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters $\varepsilon$ and $\rho$ independently tending to zero. It is assumed that $\varepsilon/\rho\to 0$. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of $\varepsilon$ and $\rho$. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio $\rho/\varepsilon$. The coefficients of the inner expansion are found from a recurrence chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the inner space variable is found.

Détail

Mots-clés : parabolic equation
Keywords: Cauchy problem, asymptotics.

Bibliographie

[1] Arnold V.I., Osobennosti kaustik i volnovykh frontov, Fazis, M., 1996, 334 pp. | MR

[2] Suleimanov B.I., “O reshenii uravneniya Kortevega - de Vriza, voznikayuschego vblizi tochki oprokidyvaniya v zadachakh s maloi dispersiei”, Pisma v zhurn. eksperiment. i teoret. fiziki, 58:11 (1993), 906–910 | MR

[3] Dubrovin B., Elaeva M., “On the critical behavior in nonlinear evolutionary PDEs with small viscosity”, Russ. J. Math. Phys., 19:4 (2012), 449–460 | DOI | MR | Zbl

[4] Garifullin R.N., Suleimanov B.I., “Ot slabykh razryvov k bezdissipativnym udarnym volnam”, Zhurn. eksperiment. i teoret. fiziki, 137:1 (2010), 149–164

[5] Teodorovich E.V., “Metod renormalizatsionnoi gruppy v zadachakh mekhaniki”, Prikl. matematika i mekhanika, 68:2 (2004), 335–367 | MR | Zbl

[6] B. Dubrovin, T. Grava, C. Klein, A. Moro, “On critical vehaviour in systems of Hamiltonian partial differential equations”, J. Nonlinear. Sci., 25:3 (2015), 631–707 | DOI | MR | Zbl

[7] Zakharov S.V., “Singulyarnye asimptotiki v zadache Koshi dlya parabolicheskogo uravneniya s malym parametrom”, Tr. In-ta matematiki i mekhaniki UrO RAN, 21:1 (2015), 97–104 | MR

[8] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[9] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977, 624 pp.

[10] Gurbatov S.N., Malakhov A.N., Saichev A.I., Nelineinye sluchainye volny v sredakh bez dispersii, Nauka, M., 1990, 216 pp. | MR

[11] Kudashev V.R., Suleimanov B.I., “Vliyanie maloi dissipatsii na protsessy zarozhdeniya odnomernykh udarnykh voln”, Prikl. matematika i mekhanika, 65:3 (2001), 456–466 | MR | Zbl

[12] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[13] Ilin A.M., “Ob asimptotike reshenii odnoi zadachi s malym parametrom”, Izv. AN SSSR. Ser. matematicheskaya, 53:2 (1989), 258–275 | MR

[14] Zakharov S.V., “Zadacha Koshi dlya kvazilineinogo parabolicheskogo uravneniya s dvumya malymi parametrami”, Dokl. Akad. Nauk, 422:6 (2008), 733–734 | Zbl

[15] Zakharov S.V., “Two-parameter asymptotics in the Cauchy problem for a quasi-linear parabolic equation”, Asympt. Analysis, 63:1–2 (2009), 49–54 | DOI | MR | Zbl

[16] Zakharov S.V., “Zadacha Koshi dlya kvazilineinogo parabolicheskogo uravneniya s bolshim nachalnym gradientom i maloi vyazkostyu”, Zhurn. vychisl. matem. i matem. fiziki, 50:4 (2010), 699–706 | MR | Zbl

[17] Godunov S.K., Uravneniya matematicheskoi fiziki, Nauka, M., 1971, 416 pp. | MR

[18] Oleinik O.A., “Razryvnye resheniya nelineinykh differentsialnykh uravnenii”, Uspekhi mat. nauk, 12:3(75) (1957), 3–73 | MR | Zbl

[19] Zakharov S.V., “O raspredelenii tepla v beskonechnom sterzhne”, Mat. zametki, 80:3 (2006), 379–385 | DOI | MR | Zbl

[20] Danilin A.R., “Asimptotika optimalnogo znacheniya funktsionala kachestva pri bystrostabiliziruyuschemsya nepryamom upravlenii v singulyarnom sluchae”, Zhurn. vychisl. matematiki i mat. fiziki, 46:12 (2006), 2166–2177 | MR

[21] Ilin A.M., Danilin A.R., Asimptoticheskie metody v analize, Fizmatlit, M., 2009, 248 pp.