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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 |
References:
[1] |
L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.
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, Lecture Notes in Math., 1927, Springer, Berlin, 2008, 1-41.
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, Lecture Notes of the Unione Matematica Italiana, 5, Springer, Berlin, 2008, 3-57.
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), 1191-1244.
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), 235-282.
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), 1237-1272.
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), 15-46.
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), 511-547.
doi: 10.1007/BF01393835. |
[9] |
D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch, Phys. Fluids, 13 (2001), 677-691.
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), 685-703.
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), 596-599 (electronic).
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), 1138-1163.
doi: 10.1137/080737629. |
[13] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, {27}, Cambridge University Press, Cambridge, 2002. |
[14] |
C. Marchioro and M. Pulvirenti, On the vortex-wave system, in Mechanics, Analysis, and Geometry: 200 Years after Lagrange, Elsevier Science, Amsterdam, 1991, 79-95. |
[15] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, New York, 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), 137-149. |
[17] |
D. Schecter, Two-dimensional vortex dynamics with background vorticity, in CP606, Non-Neutral Plasma Physics IV, Vol. 606, American Institute of Physics, 2002, 443-452.
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), 1704-1723.
doi: 10.1063/1.1359763. |
[19] |
E. Stein, Harmonic Analysis, Princeton University Press, 1993. |
show all references
References:
[1] |
L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.
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, Lecture Notes in Math., 1927, Springer, Berlin, 2008, 1-41.
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, Lecture Notes of the Unione Matematica Italiana, 5, Springer, Berlin, 2008, 3-57.
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), 1191-1244.
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), 235-282.
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), 1237-1272.
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), 15-46.
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), 511-547.
doi: 10.1007/BF01393835. |
[9] |
D. Jin and D. Dubin, Point vortex dynamics within a background vorticity patch, Phys. Fluids, 13 (2001), 677-691.
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), 685-703.
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), 596-599 (electronic).
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), 1138-1163.
doi: 10.1137/080737629. |
[13] |
A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, {27}, Cambridge University Press, Cambridge, 2002. |
[14] |
C. Marchioro and M. Pulvirenti, On the vortex-wave system, in Mechanics, Analysis, and Geometry: 200 Years after Lagrange, Elsevier Science, Amsterdam, 1991, 79-95. |
[15] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, New York, 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), 137-149. |
[17] |
D. Schecter, Two-dimensional vortex dynamics with background vorticity, in CP606, Non-Neutral Plasma Physics IV, Vol. 606, American Institute of Physics, 2002, 443-452.
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), 1704-1723.
doi: 10.1063/1.1359763. |
[19] |
E. Stein, Harmonic Analysis, Princeton University Press, 1993. |
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