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May  2016, 36(5): 2405-2417. doi: 10.3934/dcds.2016.36.2405

Flows of vector fields with point singularities and the vortex-wave system

1. 

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

2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brazil, Brazil

3. 

Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz, 91128 Palaiseau, France

Received  April 2015 Revised  August 2015 Published  October 2015

The vortex-wave system is a version of the vorticity equation governing the motion of 2D incompressible fluids in which vorticity is split into a finite sum of Diracs, evolved through an ODE, plus an $L^p$ part, evolved through an active scalar transport equation. Existence of a weak solution for this system was recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for $p>2$, but their result left open the existence and basic properties of the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the) linear transport problem associated to the vortex-wave system. To this end, we study the flow associated to a two-dimensional vector field which is singular at a moving point. We first observe that existence and uniqueness of the regular Lagrangian flow are ensured by combining previous results by Ambrosio and by Lacave and Miot. In addition we prove that, generically, the Lagrangian trajectories do not collide with the point singularity. In the second part we present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector fields.
Citation: Gianluca Crippa, Milton C. Lopes Filho, Evelyne Miot, Helena J. Nussenzveig Lopes. Flows of vector fields with point singularities and the vortex-wave system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405
References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,, Invent. Math., 158 (2004), 227.  doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields,, in Calculus of Variations and Nonlinear Partial Differential Equations, (1927), 1.  doi: 10.1007/978-3-540-75914-0_1.  Google Scholar

[3]

L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields,, in Transport Equations and Multi-D Hyperbolic Conservation Laws, (2008), 3.  doi: 10.1007/978-3-540-76781-7_1.  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]

F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral,, J. Hyper. Differential Equations, 10 (2013), 235.  doi: 10.1142/S0219891613500100.  Google Scholar

[6]

S. Caprino, C. Marchioro, E. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-d,, Comm. Partial Differential Equations, 37 (2012), 1237.  doi: 10.1080/03605302.2011.653032.  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]

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

[9]

D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch,, Phys. Fluids, 13 (2001), 677.  doi: 10.1063/1.1343484.  Google Scholar

[10]

M. C. Lopes Filho, E. Miot and H. J. Nussenzveig Lopes, Existence of a weak solution in $L^p$ to the vortex-wave system,, J. Nonlinear Science, 21 (2011), 685.  doi: 10.1007/s00332-011-9097-y.  Google Scholar

[11]

M. C. Lopes Filho and H. J. Nussenzveig Lopes, An extension of Marchioro's bound on the growth of a vortex patch to flows with $L^p$ vorticity,, SIAM J. Math. Anal., 29 (1998), 596.  doi: 10.1137/S0036141096310910.  Google Scholar

[12]

C. Lacave and E. Miot, Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex,, SIAM J. Math. Anal., 41 (2009), 1138.  doi: 10.1137/080737629.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[14]

C. Marchioro and M. Pulvirenti, On the vortex-wave system,, in Mechanics, (1991), 79.   Google Scholar

[15]

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

[16]

P. Newton, The N-vortex problem on a sphere: Geophysical mechanisms that break integrability,, Theor. Comput. Fluid Dyn., 24 (2010), 137.   Google Scholar

[17]

D. Schecter, Two-dimensional vortex dynamics with background vorticity,, in CP606, (2002), 443.  doi: 10.1063/1.1454315.  Google Scholar

[18]

D. Schecter and D. Dubin, Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient,, Phys. Fluids, 13 (2001), 1704.  doi: 10.1063/1.1359763.  Google Scholar

[19]

E. Stein, Harmonic Analysis,, Princeton University Press, (1993).   Google Scholar

show all references

References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,, Invent. Math., 158 (2004), 227.  doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields,, in Calculus of Variations and Nonlinear Partial Differential Equations, (1927), 1.  doi: 10.1007/978-3-540-75914-0_1.  Google Scholar

[3]

L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields,, in Transport Equations and Multi-D Hyperbolic Conservation Laws, (2008), 3.  doi: 10.1007/978-3-540-76781-7_1.  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]

F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral,, J. Hyper. Differential Equations, 10 (2013), 235.  doi: 10.1142/S0219891613500100.  Google Scholar

[6]

S. Caprino, C. Marchioro, E. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-d,, Comm. Partial Differential Equations, 37 (2012), 1237.  doi: 10.1080/03605302.2011.653032.  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]

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

[9]

D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch,, Phys. Fluids, 13 (2001), 677.  doi: 10.1063/1.1343484.  Google Scholar

[10]

M. C. Lopes Filho, E. Miot and H. J. Nussenzveig Lopes, Existence of a weak solution in $L^p$ to the vortex-wave system,, J. Nonlinear Science, 21 (2011), 685.  doi: 10.1007/s00332-011-9097-y.  Google Scholar

[11]

M. C. Lopes Filho and H. J. Nussenzveig Lopes, An extension of Marchioro's bound on the growth of a vortex patch to flows with $L^p$ vorticity,, SIAM J. Math. Anal., 29 (1998), 596.  doi: 10.1137/S0036141096310910.  Google Scholar

[12]

C. Lacave and E. Miot, Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex,, SIAM J. Math. Anal., 41 (2009), 1138.  doi: 10.1137/080737629.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[14]

C. Marchioro and M. Pulvirenti, On the vortex-wave system,, in Mechanics, (1991), 79.   Google Scholar

[15]

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

[16]

P. Newton, The N-vortex problem on a sphere: Geophysical mechanisms that break integrability,, Theor. Comput. Fluid Dyn., 24 (2010), 137.   Google Scholar

[17]

D. Schecter, Two-dimensional vortex dynamics with background vorticity,, in CP606, (2002), 443.  doi: 10.1063/1.1454315.  Google Scholar

[18]

D. Schecter and D. Dubin, Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient,, Phys. Fluids, 13 (2001), 1704.  doi: 10.1063/1.1359763.  Google Scholar

[19]

E. Stein, Harmonic Analysis,, Princeton University Press, (1993).   Google Scholar

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