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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

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