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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. [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. [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. [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. [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. [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. [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. [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. [9] D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch,, Phys. Fluids, 13 (2001), 677. doi: 10.1063/1.1343484. [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. [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. [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. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Applied Mathematics, (2002). [14] C. Marchioro and M. Pulvirenti, On the vortex-wave system,, in Mechanics, (1991), 79. [15] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-4284-0. [16] P. Newton, The N-vortex problem on a sphere: Geophysical mechanisms that break integrability,, Theor. Comput. Fluid Dyn., 24 (2010), 137. [17] D. Schecter, Two-dimensional vortex dynamics with background vorticity,, in CP606, (2002), 443. doi: 10.1063/1.1454315. [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. [19] E. Stein, Harmonic Analysis,, Princeton University Press, (1993).

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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. [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. [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. [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. [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. [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. [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. [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. [9] D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch,, Phys. Fluids, 13 (2001), 677. doi: 10.1063/1.1343484. [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. [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. [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. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Texts in Applied Mathematics, (2002). [14] C. Marchioro and M. Pulvirenti, On the vortex-wave system,, in Mechanics, (1991), 79. [15] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-4284-0. [16] P. Newton, The N-vortex problem on a sphere: Geophysical mechanisms that break integrability,, Theor. Comput. Fluid Dyn., 24 (2010), 137. [17] D. Schecter, Two-dimensional vortex dynamics with background vorticity,, in CP606, (2002), 443. doi: 10.1063/1.1454315. [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. [19] E. Stein, Harmonic Analysis,, Princeton University Press, (1993).
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