Exponential stability of the flow for~a~generalized Burgers equation on~a~circle

A. Djurdjevac ; A. R. Shirikyan

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 588-598.

Résumé

The paper deals with the problem of stability for the flow of the $\mathrm{1D}$ Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on $\mathbb{R}$, which is exponentially stable in $H^1$ as $t\to+\infty$. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability $1$.

Détail

Keywords: Burgers equation, exponential stability, bounded trajectory.

Bibliographie

[1] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR

[2] Kruzhkov S. N., “O zadache Koshi dlya nekotorykh klassov kvazilineinykh parabolicheskikh uravnenii”, Mat. zametki, 6:3 (1969), 295–300 | Zbl

[3] Krylov N. V., Nelineinye ellipticheskie i parabolicheskie uravneniya vtorogo poryadka, Nauka, M., 1985 | MR

[4] Krylov N. V., Safonov M. V., “Nekotoroe svoistvo reshenii parabolicheskikh uravnenii s izmerimymi koeffitsientami”, Izv. AN SSSR. Ser. mat., 44:1 (1980), 161–175 | MR | Zbl

[5] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971 | MR

[6] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972

[7] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, v. 1, Mir, M., 1971

[8] Bakhtin Y., Li L., “Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation”, Commun. Pure Appl. Math.., 72:3 (2019), 536–619 | DOI | MR | Zbl

[9] Boritchev A., “Sharp estimates for turbulence in white-forced generalised Burgers equation”, Geom. Funct. Anal., 23:6 (2013), 1730–1771 | DOI | MR | Zbl

[10] Chung J., Kwon O., “Asymptotic behavior for the viscous Burgers equation with a stationary source”, J. Math. Phys., 57:10 (2016), 101506 | DOI | MR | Zbl

[11] Dunlap A., Graham C., Ryzhik L., “Stationary solutions to the stochastic Burgers equation on the line”, Commun. Math. Phys., 382:2 (2021), 875–949 | DOI | MR | Zbl

[12] Djurdjevac A., Rosati T., Synchronisation for scalar conservation laws via Dirichlet boundary, 2022, arXiv: 2211.05814 | Zbl

[13] Djurdjevac A., Shirikyan A., Stabilisation of a viscous conservation law by a one-dimensional external force, 2022, arXiv: 2204.03427 | Zbl

[14] Evans L. C., Partial differential equations, Am. Math. Soc., Providence, 2010 | MR | Zbl

[15] Hill A. T., Süli E., “Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains”, Proc. Roy. Soc. Edinburgh Sect. A, 125:2 (1995), 439–448 | DOI | MR | Zbl

[16] Hörmander L., Lectures on nonlinear hyperbolic differential equations, Springer, Berlin, 1997 | MR | Zbl

[17] Jauslin H. R., Kreiss H. O., Moser J., “On the forced Burgers equation with periodic boundary conditions”, Differential equations (La Pietra, 1996), Am. Math. Soc., Providence, 1999, 133–153 | MR | Zbl

[18] Kalita P., Zgliczyński P., “On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions”, Proc. Roy. Soc. Edinburgh Sect. A, 150:4 (2020), 2025–2054 | DOI | MR | Zbl

[19] Kifer Y., “The Burgers equation with a random force and a general model for directed polymers in random environments”, Probab. Theory Related Fields, 108:1 (1997), 29–65 | DOI | MR | Zbl

[20] Shirikyan A., “Global exponential stabilisation for the Burgers equation with localised control”, J. Éc. Polytech. Math., 4 (2017), 613–632 | DOI | MR | Zbl

[21] Sinaĭ Ya. G., “Two results concerning asymptotic behavior of solutions of the Burgers equation with force”, J. Stat. Phys., 64:1-2 (1991), 1–12 | DOI | MR | Zbl