## Global solutions of two-dimensional Navier-Stokes and Euler equations.(English)Zbl 0837.35110

The Cauchy problem for the Navier-Stokes system in $$\mathbb{R}^2$$ is considered in the vorticity formulation. A new approach to the existence and uniqueness results based upon comparison principles for linear parabolic equations is proposed. The existence of global solutions to Navier-Stokes (NS) and Euler (E) equations with the initial vorticity in $$L^1(\mathbb{R}^2)$$ for (NS) and in $$(L^1\cap L^r)(\mathbb{R}^2)$$, $$r> 2$$, is proved. The solution to (NS) is shown to be unique, smooth and continuously dependent on initial data, and the velocity solution to (E) is Hölder continuous in the space and time coordinates. Another result is the existence of a subsequence of solutions to (NS) converging to a solution of (E) as the viscosity vanishes.

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76B47 Vortex flows for incompressible inviscid fluids
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### References:

 [1] C. Bardos, Existence et unicit? de la solution de l’?quation d’Euler en dimension deux, J. Math. Anal. Appl. 40 (1972), 769-790. · Zbl 0249.35070 [2] M. Ben-Artzi, Global existence and decay for a nonlinear parabolic equation, Nonlinear Analysis, TMA 19 (1992), 763-768. · Zbl 0782.35009 [3] M. Ben-Artzi, J. Goodman & A. Levy, Remarks on a nonlinear parabolic equation (in preparation). · Zbl 0932.35115 [4] E. A. Carlen & M. Loss, Sharp constant in Nash’s inequality, Duke Math., J. 71 (1993), 213-215. · Zbl 0822.35018 [5] A. Chorin & J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 1979. · Zbl 0417.76002 [6] R. J. DiPerna & A. Majda, Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math. 40 (1987), 301-345. · Zbl 0850.76730 [7] E. B. Fabes & D. W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338. · Zbl 0652.35052 [8] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964. · Zbl 0144.34903 [9] Y. Giga, T. Miyakawa & H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal. 104 (1988), 223-250. · Zbl 0666.76052 [10] K. K. Golovkin, Vanishing viscosity in Cauchy’s problem for hydromechanics equations, in Proceedings of the Steklov Institute of Mathematics, vol. 92 (O. A. Ladyzenskaja, ed.), 1966 (Amer. Math. Soc. translation, 1968). · Zbl 0168.35504 [11] S. Kaniel & M. Shinbrot, Smoothness of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 24 (1967), 302-324. · Zbl 0152.44902 [12] T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188-200. · Zbl 0166.45302 [13] T. Kato, Remarks on the Euler and Navier-Stokes Equations in ?2, Proc. Symp. Pure. Math. 45 (1986), Part 2, Amer. Math. Soc. 1-7. · Zbl 0598.35093 [14] T. Kato, The Navier-Stokes equation for an incompressible fluid in ?2 with a measure as the initial vorticity Diff. Integ. Eqs. 7 (1994), 949-966. · Zbl 0826.35094 [15] T. Kato & G. Ponce, Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces L s p (?2), Revista Mat. Iberoamericana 2 (1986), 73-88. · Zbl 0615.35078 [16] T. Kato & G. Ponce, On nonstationary flows of viscous and ideal fluids in l- s p (?2), Duke Math. J. 55 (1987), 487-499. · Zbl 0649.76011 [17] T. Kato & G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. · Zbl 0671.35066 [18] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969 (English translation). · Zbl 0184.52603 [19] O. A. Ladyzhenskaya, V. A. Solonnikov & N. N. Ural’eceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., 1968. [20] J. Leray, ?tude de diverses ?quations int?grales non lin?aires et des quelques probl?mes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82. · Zbl 0006.16702 [21] J. L. Lions & G. Prodi, Un th?or?me d’existence et d’unicit? dans les ?quations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris 248 (1959), 3519-3521. · Zbl 0091.42105 [22] C. Marchioro & M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys. 84 (1982), 483-503. · Zbl 0527.76021 [23] V. G. Maz’ja, Sobolev Spaces, Springer-Verlag, 1985. [24] F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1968), 328-348. · Zbl 0187.49508 [25] G. Ponce, On two dimensional incompressible fluids, Comm. Part. Diff. Eqs. 11 (1986), 483-511. · Zbl 0594.35077 [26] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195. · Zbl 0106.18302 [27] R. Temam, Navier-Stokes Equations, North-Holland, 1979. [28] W. Wolibner, Un theor?me sur l’existence du movement plan d’un fluide parfait, homog?ne, incompressible, pendant un temps infiniment long, Math. Z. 37 (1993), 698-726. · JFM 59.1447.02
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