×

Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations. (English) Zbl 1347.35041

Summary: We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to the Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the viscous Burgers’ equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish the existence of a locally attracting solution.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
35B41 Attractors
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnold, V.; Kozlov, V.; Neishtadt, A., Mathematical Aspects of Classical and Celestial Mechanics (1997), Springer Verlag: Springer Verlag Berlin · Zbl 0885.70001
[2] Bambusi, D., An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Henri Poincaré, 4, 685-712 (2003) · Zbl 1031.37056
[3] Bambusi, D., Galerkin averaging method and Poincaré normal form for some quasilinear PDEs, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4, 4, 669-702 (2005) · Zbl 1170.35317
[4] Bogolyubov, N. N.; Mitropol’skii, Y. A., Asymptotic Methods in the Theory of Nonlinear Oscillations (1961), Gordon & Breach: Gordon & Breach New York · Zbl 0151.12201
[5] Bogolyubov, N. N.; Zubarev, D. N., An asymptotic approximation method for a system with rotating phases and its application to the motion of a charged particle in a magnetic field, Ukraïn. Mat. Zh., 7 (1955) · Zbl 0068.39601
[6] Burgers, J. M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1, 171-199 (1948)
[7] Ćwiszewski, A., Averaging principle and hyperbolic evolution equations, Nonlinear Anal., 75, 2362-2375 (2012) · Zbl 1253.47050
[8] Cyranka, J., Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof, Topol. Methods Nonlinear Anal., 45, 655-697 (2015) · Zbl 1365.65220
[9] Cyranka, J.; Zgliczyński, P., Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof, SIAM J. Appl. Dyn. Syst., 14, 787-821 (2015) · Zbl 1312.65170
[10] Cyranka, J.; Mucha, P. B.; Titi, E. S.; Zgliczyński, P., Stabilizing the long-time behavior of the Navier-Stokes equations and damped Euler systems by fast oscillating forces (2016), preprint
[11] Dahlquist, G., Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations (1958), Almqvist & Wiksells: Almqvist & Wiksells Uppsala, Transactions of the Royal Institute of Technology, Stockholm, 1959
[12] Foias, C.; Manley, O.; Rosa, R.; Temam, R., Navier-Stokes Equations and Turbulence, Encyclopedia Math. Appl., vol. 84 (2008), Cambridge University Press · Zbl 1139.35001
[13] Freidlin, M.; Wentzell, A., Random Perturbations of Dynamical Systems, Grundlehren Math. Wiss., vol. 260 (2012), Springer · Zbl 1267.60004
[14] Gerschgorin, S., Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR Ser. Fiz.-Mat., 6, 749-754 (1931) · Zbl 0003.00102
[15] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I, Nonstiff Problems (1987), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0638.65058
[16] Hale, J. K.; Verduyn Lunel, S. M., Averaging in infinite dimensions, J. Integral Equations Appl., 2, 463-494 (1990) · Zbl 0755.45012
[17] Henry, D., (Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840 (1981), Springer) · Zbl 0456.35001
[18] Iserles, A., Three stories of high oscillation, Eur. Math. Soc. Newsl., 87 (2013) · Zbl 1360.65189
[19] Jauslin, H. R.; Kreiss, H. O.; Moser, J., On the forced Burgers equation with periodic boundary condition, Proc. Sympos. Pure Math., 65 (1999) · Zbl 0930.35156
[20] Kapela, T.; Zgliczyński, P., A Lohner-type algorithm for control systems and ordinary differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 11, 365-385 (2009) · Zbl 1185.65079
[21] Lozinskii, S. M., Error estimates for the numerical integration of ordinary differential equations, part I, Izv. Vyssh. Uchebn. Zaved. Mat., 6, 52-90 (1958), (Russian) · Zbl 0198.21202
[22] Matthies, K., Time-averaging under fast periodic forcing of parabolic partial differential equations: exponential estimates, J. Differential Equations, 174, 133-180 (2001) · Zbl 1023.35055
[23] Matthies, K., Homogenisation of exponential order for elliptic systems in infinite cylinders, Asymptot. Anal., 43, 205-232 (2005) · Zbl 1077.35021
[24] Matthies, K., Exponential averaging under rapid quasiperiodic forcing, Adv. Differential Equations, 13, 427-456 (2008) · Zbl 1154.37371
[25] Matthies, K.; Scheel, A., Exponential averaging of Hamiltonian evolution equations, Trans. Amer. Math. Soc., 355, 747-773 (2003) · Zbl 1008.37043
[26] Mattingly, J.; Sinai, Y., An elementary proof of the existence and uniqueness theorem for Navier-Stokes equations, Commun. Contemp. Math., 1, 4, 497-516 (1999) · Zbl 0961.35112
[27] Narkiewicz, W., Classical Problems in Number Theory (1986), PWN-Polish Scientific Publishers: PWN-Polish Scientific Publishers Warszawa · Zbl 0593.10035
[28] Neishtadt, A., The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48, 133-139 (1984) · Zbl 0571.70022
[29] Prizzi, M., Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations, J. Differential Equations, 210, 429-451 (2005) · Zbl 1065.37017
[30] Sinai, Y., Navier-Stokes system with periodic boundary conditions, Regul. Chaotic Dyn., 4, 2, 3-15 (1999) · Zbl 1005.35074
[31] Ward, J. R., Homotopy and bounded solutions of ordinary differential equations, J. Differential Equations, 107, 428-445 (1994) · Zbl 0794.34027
[32] Weinan, W. E.; Sinai, Y., New results in mathematical and statistical hydrodynamics, Uspekhi Mat. Nauk, 55, 4(334), 25-58 (2000), (Russian) · Zbl 0983.76010
[33] Whitham, G. B., Linear and Nonlinear Waves (1975), John Wiley & Sons · Zbl 0373.76001
[34] Zgliczyński, P., Trapping regions and an ODE-type proof of an existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Univ. Iagel. Acta Math., 41, 89-113 (2003) · Zbl 1067.35068
[35] Zgliczyński, P., On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach, J. Differential Equations, 195, 271-283 (2003) · Zbl 1043.35034
[36] Zgliczyński, P., Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE - a computer assisted proof, Found. Comput. Math., 4, 157-185 (2004) · Zbl 1066.65105
[37] Zgliczyński, P., Rigorous numerics for dissipative PDEs III. An effective algorithm for rigorous integration of dissipative PDEs, Topol. Methods Nonlinear Anal., 36, 197-262 (2010) · Zbl 1230.65113
[38] Zgliczyński, P.; Mischaikow, K., Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math., 1, 255-288 (2001) · Zbl 0984.65101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.