December  2015, 8(4): 685-689. doi: 10.3934/krm.2015.8.685

Strong continuity for the 2D Euler equations

1. 

Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

2. 

Department of Epidemiology and Public Health, Swiss Tropical and Public Health Institute, Universität Basel, Socinstrasse 57, 4051 Basel, Switzerland

3. 

GSSI - Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 L'Aquila, Italy

Received  April 2015 Revised  May 2015 Published  July 2015

We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.
Citation: Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685
References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps: Results and counterexamples,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 863. Google Scholar

[2]

G. Alberti, S. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions,, J. Eur. Math. Soc. (JEMS), 16 (2014), 201. doi: 10.4171/JEMS/431. Google Scholar

[3]

G. Alberti, S. Bianchini and G. Crippa, On the $L^p$ differentiability of certain classes of functions,, Revista Matemática Iberoamericana, 30 (2014), 349. doi: 10.4171/RMI/782. Google Scholar

[4]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1191. doi: 10.1017/S0308210513000085. Google Scholar

[5]

A. Bohun, F. Bouchut and G. Crippa, Lagrangian solutions to the Euler equations with $L^1$ vorticity and infinite energy,, In preparation, (2015). Google Scholar

[6]

F. Bouchut and G. Crippa, Transport equations with coefficient having a gradient given by a singular integral and applications,, J. Hyper. Differential Equations, 10 (2013), 235. doi: 10.1142/S0219891613500100. Google Scholar

[7]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow,, J. Reine Angew. Math., 616 (2008), 15. doi: 10.1515/CRELLE.2008.016. Google Scholar

[8]

G. Crippa and S. Spirito, Renormalized solutions of the 2d Euler equations,, Comm. Math. Phys., 339 (2015), 191. doi: 10.1007/s00220-015-2411-z. Google Scholar

[9]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[10]

R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow,, Comm. Pure Appl. Math., 40 (1987), 301. doi: 10.1002/cpa.3160400304. Google Scholar

[11]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density,, J. Math. Pures Appl., 86 (2006), 68. doi: 10.1016/j.matpur.2006.01.005. Google Scholar

[12]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Appl. Math, (2002). Google Scholar

[13]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-4284-0. Google Scholar

[14]

T. C. Sideris and L. Vega, Stability in $L^1$ of circular vortex patches,, Proceedings of the AMS, 137 (2009), 4199. doi: 10.1090/S0002-9939-09-10048-5. Google Scholar

[15]

Y. Tang, Nonlinear stability of vortex patches,, Transactions of the AMS, 304 (1987), 617. doi: 10.1090/S0002-9947-1987-0911087-X. Google Scholar

[16]

Y. H. Wan, The stability of rotating vortex patches,, Comm. Math. Phys., 107 (1986), 1. doi: 10.1007/BF01206950. Google Scholar

[17]

Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches,, Comm. Math. Phys., 99 (1985), 435. doi: 10.1007/BF01240356. Google Scholar

[18]

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

show all references

References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps: Results and counterexamples,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 863. Google Scholar

[2]

G. Alberti, S. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions,, J. Eur. Math. Soc. (JEMS), 16 (2014), 201. doi: 10.4171/JEMS/431. Google Scholar

[3]

G. Alberti, S. Bianchini and G. Crippa, On the $L^p$ differentiability of certain classes of functions,, Revista Matemática Iberoamericana, 30 (2014), 349. doi: 10.4171/RMI/782. Google Scholar

[4]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1191. doi: 10.1017/S0308210513000085. Google Scholar

[5]

A. Bohun, F. Bouchut and G. Crippa, Lagrangian solutions to the Euler equations with $L^1$ vorticity and infinite energy,, In preparation, (2015). Google Scholar

[6]

F. Bouchut and G. Crippa, Transport equations with coefficient having a gradient given by a singular integral and applications,, J. Hyper. Differential Equations, 10 (2013), 235. doi: 10.1142/S0219891613500100. Google Scholar

[7]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow,, J. Reine Angew. Math., 616 (2008), 15. doi: 10.1515/CRELLE.2008.016. Google Scholar

[8]

G. Crippa and S. Spirito, Renormalized solutions of the 2d Euler equations,, Comm. Math. Phys., 339 (2015), 191. doi: 10.1007/s00220-015-2411-z. Google Scholar

[9]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[10]

R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow,, Comm. Pure Appl. Math., 40 (1987), 301. doi: 10.1002/cpa.3160400304. Google Scholar

[11]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density,, J. Math. Pures Appl., 86 (2006), 68. doi: 10.1016/j.matpur.2006.01.005. Google Scholar

[12]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Appl. Math, (2002). Google Scholar

[13]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-4284-0. Google Scholar

[14]

T. C. Sideris and L. Vega, Stability in $L^1$ of circular vortex patches,, Proceedings of the AMS, 137 (2009), 4199. doi: 10.1090/S0002-9939-09-10048-5. Google Scholar

[15]

Y. Tang, Nonlinear stability of vortex patches,, Transactions of the AMS, 304 (1987), 617. doi: 10.1090/S0002-9947-1987-0911087-X. Google Scholar

[16]

Y. H. Wan, The stability of rotating vortex patches,, Comm. Math. Phys., 107 (1986), 1. doi: 10.1007/BF01206950. Google Scholar

[17]

Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches,, Comm. Math. Phys., 99 (1985), 435. doi: 10.1007/BF01240356. Google Scholar

[18]

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

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