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On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

Part of: Parabolic equations and systems Infinite-dimensional dissipative dynamical systems Partial differential equations

Published online by Cambridge University Press:  14 March 2019

Piotr Kalita  and
Piotr Zgliczyński
Piotr Kalita
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland (piotr.kalita@ii.uj.edu.pl; piotr.zgliczynski@ii.uj.edu.pl)
Piotr Zgliczyński
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland (piotr.kalita@ii.uj.edu.pl; piotr.zgliczynski@ii.uj.edu.pl)
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Abstract

We study the non-autonomously forced Burgers equation

$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$
on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

Keywords

Burgers equationnon-autonomous forcingasymptotic behaviourglobal stabilityeternal solutionenergy methodmaximum principlepullback attractor

MSC classification

Primary: 37L05: General theory, nonlinear semigroups, evolution equations
Secondary: 35B35: Stability 35B41: Attractors 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 2025 - 2054
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Babin, A. V. and Vishik, M. I.. Attractors of evolution equations (Amsterdam, London, New York, Tokyo: North Holland, 1992).Google Scholar
2Balibrea, F., Caraballo, T., Kloeden, P. E. and Valero, J.. Recent developments in dynamical systems: three perspectives. Int. J. Bifurcat. Chaos 20 (2010), 2591–2636.CrossRefGoogle Scholar
3Ball, J. M.. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Sci. 7 (1997), 475–502, Erratum, ibid 8 (1998) 233, corrected version appears in “Mechanics: from Theory to Computation”, 447–474, Springer Verlag, 2000.CrossRefGoogle Scholar
4Byrnes, C. I., Gilliam, D. S., Shubov, V. I. and Xu, Z.. Steady state resonse to Burgers' equation with varying viscosity. In Computation and Control IV, Progress in Systems and Control Theory, vol. 20, Birkhäuser, 1995, 75–97.CrossRefGoogle Scholar
5Cao, C. and Titi, E. S.. Asymptotic behavior of viscous 1-D scalar conservation laws with Neumann boundary conditions. In Mathematics and mathematics education, Bethlehem, 2000 (eds. Elaydi, S., Titi, E. S., Saleh, M., Jain, S. K., and Abu Saris, R.) (River Edge, NJ: World Science Publications, 2002), 306–324.Google Scholar
6Caraballo, T., Łukaszewicz, G. and Real, J.. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal. Theory Methods Appl. 64(3), 484–498.CrossRefGoogle Scholar
7Carvalho, A. N., Langa, J. A. and Robinson, J. C.. Attractors for infinite-dimensional non-autonomous dynamical systems, Applied Mathematical Series vol. 182, (New York: Springer, 2013).CrossRefGoogle Scholar
8Cheban, D. N., Kloeden, P. E. and Schmalfuss, B.. Pullback attractors in dissipative non-autonomous differential equations under discretization. J. Dyn. Differ. Equ. 13 (2001), 185–213.CrossRefGoogle Scholar
9Chen, S.-H., Hsia, C.-H., Jung, C.-Y. and Kwon, B.. Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers' equation. J. Math. Anal. Appl. 445 (2017), 655–676.CrossRefGoogle Scholar
10Cholewa, J. W. and Dłotko, T.. Bi-spaces global attractors in abstract parabolic equations, in: Banach Center Publications, vol. 60, PWN, 2003, 13–26.CrossRefGoogle Scholar
11Córdoba, A. and Córdoba, D.. A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249 (2004), 511–528.CrossRefGoogle Scholar
12Coti Zelati, M. and Kalita, P.. Minimality properties of set-valued processes and their pullback attractors. SIAM J. Math. Anal. 47 (2015), 1530–1561.CrossRefGoogle Scholar
13Crauel, H. and Flandoli, F.. Attractors for random dynamical systems. Prob. Theor. Related Fields 100 (1994), 365–393.CrossRefGoogle Scholar
14Cyranka, J.. Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof. Topol. Methods. Nonlinear. Anal. 45 (2015), 655–697.CrossRefGoogle Scholar
15Cyranka, J. and Zgliczyński, P.. Existence of globally attracting solutions for one-dimensional viscous Burgers equation with non-autonomous forcing - a computer assisted proof. SIAM J. Appl. Dynamical Syst. 14 (2015), 787–821.CrossRefGoogle Scholar
16Fontes, M. and Verdier, O.. Time-periodic solutions of the Burgers equation. J. Math. Fluid Mech. 11 (2009), 303–323.CrossRefGoogle Scholar
17Friedman, A.. Partial differential equations of parabolic type (Englewood Cliffs, N.J.: Prentice Hall, 1964).Google Scholar
18Hill, A. T. and Süli, E.. Dynamics of a nonlinear convection–diffusion equation in multidimensional bounded domains Proceedings of the Royal Society of Edinburgh 125A (1995) 439–448.CrossRefGoogle Scholar
19Jakubowski, T. and Serafin, G.. Stable estimates for source solution of critical fractal Burgers equation. Nonlinear Anal. (2016) 396–407.CrossRefGoogle Scholar
20Jauslin, H. R., Kreiss, H. O. and Moser, J.. On the forced Burgers equation with periodic boundary conditions, in: Differential Equations, La Pietra 1996, Florence, Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, 133–153.CrossRefGoogle Scholar
21Kiselev, A., Nazarov, F. and Shterenberg, R.. Blow up and regularity for fractal Burgers equation. Dyn. Partial Differ. Equ. 5 (2008), 211–240.CrossRefGoogle Scholar
22Kloeden, P. E. and Rasmussen, M.. Nonautonomous dynamical systems, Mathematical Surveys and Monographs vol. 176 (Providence, RI: Americal Mathematical Society, 2011).CrossRefGoogle Scholar
23Ly, H. V., Mease, K. D. and Titi, E. S.. Distributed and boundary control of the viscous Burgers equation. Numer. Funct. Anal. Optim. 18 (1997), 143–188.CrossRefGoogle Scholar
24Protter, M. H. and Weinberger, H. F.. Maximum principles in differential equations (New York, Berlin, Heidelberg, Tokyo: Springer-Verlag, 1984).CrossRefGoogle Scholar
25Stampacchia, G.. Contribuiti alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secundo ordere ellittiche. Ann. Scuoala Norm. Sup. Pisa Cl. Sci. (3) 12 (1958), 223–244.Google Scholar
26Temam, R.. Infinite dimensional dynamical systems in mechanics and physics, 2nd edn (New York: Springer–Verlag, 1997).CrossRefGoogle Scholar