Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$
Tóm tắt
We study the so-called damped Navier–Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier–Stokes type problems. Note that any divergent free vector field
$${u_0 \in L^\infty(\mathbb{R}^2)}$$
is allowed and no assumptions on the spatial decay of solutions as
$${|x| \to \infty}$$
are posed. In addition, applying the developed theory to the case of the classical Navier–Stokes problem in
$${\mathbb{R}^2}$$
, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.
Tài liệu tham khảo
Afendikov A., Mielke A.: Dynamical properties of spatially non-decaying 2D Navier–Stokes flows with Kolmogorov forcing in an infinite strip. J. Math. Fluid Mech. 7, 51–67 (2005)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989) (North Holland, Amsterdam, 1992)
Brull S., Pareschi L.: Dissipative hydrodynamic models for the diffusion of impurities in a gas. Appl. Math. Lett. 19, 516–521 (2006)
Chepyzhov V.V., Vishik M.I.: Trajectory attractors for dissipative 2D Euler and Navier–Stokes equations. Russian J. Math. Phys. 15(2), 156–170 (2008)
Chepyzhov V., Vishik M., Zelik S.: Strong trajectory attractors for dissipative Euler equations. J. Math. Pures Appl., (9) 96(4), 395–407 (2011)
Constantin P., Ramos F.: Inviscid limit for damped and driven incompressible Navier–Stokes equations in \({\mathbb{R}^2}\). Comm. Math. Phys. 275(2), 529–551 (2007)
Efendiev M., Miranville A., Zelik S.: Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 134(2), 271–315 (2004)
Efendiev M., Zelik S.: . Comm. Pure Appl. Math. 54(6), 625–688 (2001)
Fursikov A., Gunzburger M., Hou L.: Inhomogeneous boundary value problems for the three-dimensional evolutionary Navier–Stokes equations. J. Math. Fluid Mech. 4, 45–75 (2002)
Fursikov, A.: Flow of a viscous incompressible fluid around a body: boundary value problems and work minimization for a fluid. (Russian) Sovrem. Mat. Fundam. Napravl. 37, 83–130 (2010); translation in J. Math. Sci. (N. Y.) 180 (2012), no. 6, 763–816
Galdi G., Maremonti P., Zhou Y.: On the Navier Stokes problem in exterior domains with non decaying initial data. J. Math. Fluid Mech. 14, 633–652 (2012)
Giga Y., Matsui S., Sawada O.: Global existence of two-dimensional Navier–Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3, 302–315 (2001)
Ilyin, A.A.: The Euler equations with dissipation. Mat. Sb. 182(12), 1729–1739 (1991)[Sb. Math. 74(2), 475–485 (1993)]
Ilyin A. A., Titi E. S.: Sharp estimates for the number of degrees of freedom of the damped-driven 2D Navier–Stokes equations. J. Nonlinear Sci. 16(3), 233–253 (2006)
Ilyin A.A., Miranville A., Titi E.S.: Small viscosity sharp estimates for the global attractor of the 2D damped-driven Navier–Stokes equations. Commun. Math. Sci. 2(3), 403–426 (2004)
Lemarie-Rieusset, P.: Recent developments in the Navier–Stokes problem. In: Research Notes in Mathematics, vol. 431, Chapman & Hall, Boca Raton (2002)
Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., pp. 103–200. Elsevier, Amsterdam (2008)
Pedlosky J.: Geophysical Fluid Dynamics. Springer, New York (1979)
Pennant, J., Zelik, S.: Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in \({\mathbb{R}^3}\). Comm. Pure Appl. Anal. 12(1), 461–480 (2013)
Sawada J., Taniuchi Y.: A remark on L ∞-solutions to the 2D Navier–Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)
Temam, R.: Navier–Stokes equations, theory and numerical analysis. North-Holland, Amsterdam (1977)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. In: Applied Mathematics Series, 2nd edn. Springer, Berlin (1988) (2nd edn., New York, 1997)
Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Vishik, M.I., Fursikov, A.V.: Matematicheskie zadachi statisticheskoi gidromekhaniki. (Russian) [Mathematical problems of statistical hydromechanics] “Nauka”, Moscow, (1980) [English translation: Mathematical problems of statistical hydromechanics. Kluwer Academic publishers, Dorrend, Boston, London (1988)]
Zelik S.: Spatially nondecaying solutions of the 2D Navier–Stokes equation in a strip. Glasg. Math. J. 49(3), 525–588 (2007)
Zelik, S. Weak spatially nondecaying solutions of 3D Navier–Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II. Int. Math. Ser. (N.Y.) 7, 255–327 (2008)
Zelik S.: Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm. Pure Appl. Math. 56(5), 584–637 (2003)