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Global existence for some transport equations with nonlocal velocity. (English) Zbl 1326.35076
Summary: In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of a one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and \( n\) dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.

MSC:
35F25 Initial value problems for nonlinear first-order PDEs
35Q35 PDEs in connection with fluid mechanics
35R09 Integral partial differential equations
35D30 Weak solutions to PDEs
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References:
[1] Baker, G. R.; Li, X.; Morlet, A. C., Analytic structure of 1D transport equations with nonlocal fluxes, Phys. D, 91, 349-375, (1996) · Zbl 0899.76104
[2] Balodis, P.; Cordoba, A., An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math., 214, 1, 1-39, (2007) · Zbl 1133.35078
[3] Beale, J. T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94, 1, 61-66, (1984) · Zbl 0573.76029
[4] Berselli, L. C., Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations, Indiana Univ. Math. J., 51, 4, 905-930, (2002) · Zbl 1044.35055
[5] Berselli, L. C.; Córdoba, D.; Granero-Belinchón, R., Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., 16, 2, 175-213, (2014) · Zbl 1295.35385
[6] Bertozzi, A.; Majda, A., Vorticity and incompressible flow, (2002), Cambridge Univ. Press · Zbl 0983.76001
[7] Blanchet, A.; Carlen, E. A.; Carrillo, J. A., Functional inequalities, thick tails and asymptotics for the critical mass patlak-Keller-Segel model, J. Funct. Anal., 262, 5, 2142-2230, (2012) · Zbl 1237.35155
[8] Blanchet, A.; Dolbeault, J.; Perthame, B., Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 2006, 44, 1-32, (2006)
[9] Caffarelli, L.; Vázquez, J. L., Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202, 537-565, (2011) · Zbl 1264.76105
[10] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171, 3, 1903-1930, (2010) · Zbl 1204.35063
[11] Carrillo, J. A.; Ferreira, L. C.F.; Precioso, J. C., A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231, 1, 306-327, (2012) · Zbl 1252.35224
[12] Carrillo, J. A.; Lisini, S.; Mainini, E., Uniqueness for Keller-Segel-type chemotaxis models, Discrete Contin. Dyn. Syst., 34, 4, 1319-1338, (2014) · Zbl 1277.35009
[13] Castro, A.; Córdoba, D., Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219, 6, 1916-1936, (2008) · Zbl 1186.35002
[14] Castro, A.; Córdoba, D., Self-similar solutions for a transport equation with non-local flux, Chin. Ann. Math. Ser. B, 30, 5, 505-512, (2009) · Zbl 1186.35154
[15] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F., Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., 208, 3, 805-909, (2013) · Zbl 1293.35234
[16] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Lopez-Fernandez, M., Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175, 2, 909-948, (2012) · Zbl 1267.76033
[17] Castro, A.; Córdoba, D.; Gancedo, F.; Orive, R., Incompressible flow in porous media with fractional diffusion, Nonlinearity, 22, 8, 1791-1815, (2009) · Zbl 1169.76058
[18] Chae, D., On the transport equations with singular/regular nonlocal velocities, SIAM J. Math. Anal., 46, 2, 1017-1029, (2014) · Zbl 1321.35155
[19] Chae, D.; Constantin, P.; Córdoba, D.; Gancedo, F.; Wu, J., Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math., 65, 8, 1037-1066, (2012) · Zbl 1244.35108
[20] Chae, D.; Cordoba, A.; Cordoba, D.; Fontelos, M., Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194, 1, 203-223, (2005) · Zbl 1128.76372
[21] Constantin, P.; Córdoba, D.; Gancedo, F.; Strain, R., On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS), 15, 1, 201-227, (2013) · Zbl 1258.35002
[22] Constantin, P.; Lax, P.; Majda, A., A simple one-dimensional model for the three dimensional vorticity, Comm. Pure Appl. Math., 38, 715-724, (1985) · Zbl 0615.76029
[23] Constantin, P.; Majda, A.; Tabak, E., Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity, 7, 1495-1533, (1994)
[24] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 3, 511-528, (2004) · Zbl 1309.76026
[25] Córdoba, A.; Córdoba, D.; Fontelos, M., Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2), 162, 1-13, (2005) · Zbl 1101.35052
[26] Córdoba, D.; Gancedo, F., Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys., 273, 2, 445-471, (2007) · Zbl 1120.76064
[27] Córdoba, D.; Granero-Belinchón, R.; Orive, R., On the confined Muskat problem: differences with the deep water regime, Commun. Math. Sci., 12, 3, 423-455, (2014) · Zbl 1301.35102
[28] De Gregorio, S., On a one-dimensional model for the 3D vorticity equation, J. Stat. Phys., 59, 1251-1263, (1990) · Zbl 0712.76027
[29] Deslippe, J.; Tedstrom, R.; Daw, M. S.; Chrzan, D.; Neeraj, T.; Mills, M., Dynamics scaling in a simple one-dimensional model of dislocation activity, Philos. Mag., 84, 244-2454, (2004)
[30] Dong, H., Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, 11, 3070-3097, (2008) · Zbl 1170.35004
[31] Friedlander, S.; Gancedo, F.; Sun, W.; Vicol, V., On a singular incompressible porous media equation, J. Math. Phys., 53, 11, 115602, (2012), 20 pp · Zbl 1452.76233
[32] Friedlander, S.; Rusin, W.; Vicol, V., The magneto-geostrophic equations: a survey, (Proceedings of Advances in Mathematical Analysis of PDE, in honor of Olga Ladyzhenskaya, Amer. Math. Soc. Trasl., vol. 232, (2014)) · Zbl 1307.76027
[33] Friedlander, S.; Vicol, V., Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 2, 283-301, (2011) · Zbl 1277.35291
[34] Gancedo, F., Existence for the α-patch model and the QG sharp front in Sobolev spaces, Adv. Math., 217, 6, 2569-2598, (2008) · Zbl 1148.35099
[35] Gómez-Serrano, J.; Granero-Belinchón, R., On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity, 27, 6, 1471-1498, (2014) · Zbl 1298.76178
[36] Granero-Belinchón, R., Global existence for the confined Muskat problem, SIAM J. Math. Anal., 46, 2, 1651-1680, (2014) · Zbl 1298.35143
[37] Head, A. K., Dislocation group dynamics I. similarity solutions of the n-body problem, Philos. Mag., 26, 43-53, (1977)
[38] Head, A. K., Dislocation group dynamics II. general solutions of the n-body problem, Philos. Mag., 26, 55-63, (1977)
[39] Head, A. K., Dislocation group dynamics III. similarity solutions of the continuum approximation, Philos. Mag., 26, 65-72, (1977)
[40] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 3, 445-453, (2007) · Zbl 1121.35115
[41] Kiselev, A., Regularity and blow up for active scalars, Math. Model. Nat. Phenom., 5, 225-255, (2010) · Zbl 1194.35490
[42] Lazar, O., Global existence for the critical dissipative surface quasi-geostrophic equation, Comm. Math. Phys., 322, 1, 73-93, (2013) · Zbl 1272.35069
[43] Lazar, O., Global and local existence for the dissipative critical SQG equation with small oscillations, arXiv preprint · Zbl 1326.35278
[44] Li, D.; Rodrigo, J., Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation, Comm. Math. Phys., 286, 1, 111-124, (2009) · Zbl 1172.86301
[45] Li, D.; Rodrigo, J., Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217, 6, 2563-2568, (2008) · Zbl 1138.35381
[46] Li, D.; Rodrigo, J., On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43, 1, 507-526, (2011) · Zbl 1231.35172
[47] Lions, P. L., Mathematical topics in fluid dynamics, vol. 2. compressible models, (1998), Oxford Science Publication Oxford
[48] Marchand, F., Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces \(L^p\) or \(\dot{H}^{1 / 2}\), Comm. Math. Phys., 277, 45-67, (2008) · Zbl 1134.35025
[49] Moffatt, H. K.; Loper, D. E., The magnetostrophic rise of a buoyant parcel in the Earth’s core, Geophys. J. Int., 117, 2, 394-402, (1994)
[50] Morlet, A., Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221, 132-160, (1998) · Zbl 0916.35049
[51] Pedlosky, J., Geophysical fluid dynamics, (1986), Springer-Verlag
[52] Rodrigo, J., On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math., 58, 6, 821-866, (2005) · Zbl 1073.35006
[53] Temam, R., Navier-Stokes equations: theory and numerical analysis, (2001), AMS Chelsea Publishing Providence · Zbl 0981.35001
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