May  2019, 39(5): 2361-2392. doi: 10.3934/dcds.2019100

Well-posedness of the 2D Euler equations when velocity grows at infinity

1. 

Department of Mathematics, 368 Kidder Hall, Oregon State University, Corvallis, OR 97331, USA

2. 

Department of Mathematics, University of California, Riverside, USA

Received  September 2017 Revised  September 2018 Published  January 2019

We prove the uniqueness and finite-time existence of bounded-vorticity solutions to the 2D Euler equations having velocity growing slower than the square root of the distance from the origin, obtaining global existence for more slowly growing velocity fields. We also establish continuous dependence on initial data.

Citation: Elaine Cozzi, James P. Kelliher. Well-posedness of the 2D Euler equations when velocity grows at infinity. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2361-2392. doi: 10.3934/dcds.2019100
References:
[1]

D. M. AmbroseJ. P. KelliherM. C. Lopes Filho and H. J. Nussenzveig Lopes, Serfati solutions to the 2D Euler equations on exterior domains, J. Differential Equations, 259 (2015), 4509-4560.  doi: 10.1016/j.jde.2015.06.001.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[4]

E. Cozzi, Solutions to the 2D Euler equations with velocity unbounded at infinity, J. Math. Anal. Appl., 423 (2015), 144-161.  doi: 10.1016/j.jmaa.2014.09.053.

[5]

E. Cozzi and J. P. Kelliher, Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type, J. Differential Equations, 235 (2007), 647-657.  doi: 10.1016/j.jde.2006.12.022.

[6]

E. Cozzi and J. P. Kelliher, Incompressible Euler equations and the effect of changes at a distance, J. Math. Fluid Mech., 18 (2016), 765-781.  doi: 10.1007/s00021-016-0268-3.

[7]

T. M. Elgindi and I.-J. Jeong, Symmetries and critical phenomena in fluids, arXiv: 1610.09701v2.

[8]

T. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations, Ann. Fac. Sci. Toulouse, 26 (2017), 979-1027.  doi: 10.5802/afst.1558.

[9]

J. P. Kelliher, On the flow map for 2D Euler equations with unbounded vorticity, Nonlinearity, 24 (2011), 2599-2637.  doi: 10.1088/0951-7715/24/9/013.

[10]

J. P. Kelliher, A characterization at infinity of bounded vorticity, bounded velocity solutions to the 2D Euler equations, Indiana Univ. Math. J., 64 (2015), 1643-1666.  doi: 10.1512/iumj.2015.64.5717.

[11]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[13]

F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 27 (1967), 329-348.  doi: 10.1007/BF00251436.

[14]

P. Serfati, Solutions C en temps, n-log Lipschitz bornées en espace et équation d'Euler, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 555-558. 

[15]

Y. TaniuchiT. Tashiro and T. Yoneda, On the two-dimensional Euler equations with spatially almost periodic initial data, J. Math. Fluid Mech., 12 (2010), 594-612.  doi: 10.1007/s00021-009-0304-7.

[16]

V. I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066 (Russian).

[17]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38.  doi: 10.4310/MRL.1995.v2.n1.a4.

[18]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $ {\mathbb R} ^2$, J. Math. Fluid Mech., 15 (2013), 717-745.  doi: 10.1007/s00021-013-0144-3.

show all references

References:
[1]

D. M. AmbroseJ. P. KelliherM. C. Lopes Filho and H. J. Nussenzveig Lopes, Serfati solutions to the 2D Euler equations on exterior domains, J. Differential Equations, 259 (2015), 4509-4560.  doi: 10.1016/j.jde.2015.06.001.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J.-Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[4]

E. Cozzi, Solutions to the 2D Euler equations with velocity unbounded at infinity, J. Math. Anal. Appl., 423 (2015), 144-161.  doi: 10.1016/j.jmaa.2014.09.053.

[5]

E. Cozzi and J. P. Kelliher, Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type, J. Differential Equations, 235 (2007), 647-657.  doi: 10.1016/j.jde.2006.12.022.

[6]

E. Cozzi and J. P. Kelliher, Incompressible Euler equations and the effect of changes at a distance, J. Math. Fluid Mech., 18 (2016), 765-781.  doi: 10.1007/s00021-016-0268-3.

[7]

T. M. Elgindi and I.-J. Jeong, Symmetries and critical phenomena in fluids, arXiv: 1610.09701v2.

[8]

T. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations, Ann. Fac. Sci. Toulouse, 26 (2017), 979-1027.  doi: 10.5802/afst.1558.

[9]

J. P. Kelliher, On the flow map for 2D Euler equations with unbounded vorticity, Nonlinearity, 24 (2011), 2599-2637.  doi: 10.1088/0951-7715/24/9/013.

[10]

J. P. Kelliher, A characterization at infinity of bounded vorticity, bounded velocity solutions to the 2D Euler equations, Indiana Univ. Math. J., 64 (2015), 1643-1666.  doi: 10.1512/iumj.2015.64.5717.

[11]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[13]

F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 27 (1967), 329-348.  doi: 10.1007/BF00251436.

[14]

P. Serfati, Solutions C en temps, n-log Lipschitz bornées en espace et équation d'Euler, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 555-558. 

[15]

Y. TaniuchiT. Tashiro and T. Yoneda, On the two-dimensional Euler equations with spatially almost periodic initial data, J. Math. Fluid Mech., 12 (2010), 594-612.  doi: 10.1007/s00021-009-0304-7.

[16]

V. I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066 (Russian).

[17]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38.  doi: 10.4310/MRL.1995.v2.n1.a4.

[18]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $ {\mathbb R} ^2$, J. Math. Fluid Mech., 15 (2013), 717-745.  doi: 10.1007/s00021-013-0144-3.

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