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A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity. (English) Zbl 1252.35224
Summary: We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space \(\mathcal P_{2}(\mathbb R)\) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed by L. Ambrosio, N. Gigli and G. Savaré [Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser (2005; Zbl 1090.35002)]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in \(\mathcal P_{2}(\mathbb R)\), which was already obtained by P. Biler, G. Karch and R. Monneau [Commun. Math. Phys. 294, No. 1, 145–168 (2010; Zbl 1207.82049)]. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials.

35Q35 PDEs in connection with fluid mechanics
35C06 Self-similar solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Agueh, M., Existence of solutions to degenerate parabolic equations via the monge – kantorovich theory, Adv. differential equations, 10, 3, 309-360, (2005) · Zbl 1103.35051
[2] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient flows: in metric spaces and in the space of probability masures, (2005), Birkhäuser
[3] Baker, G.R.; Li, X.; Morlet, A.C., Analytic structure of two 1D-transport equations with nonlocal fluxes, Physica D, 91, 4, 349-375, (1996) · Zbl 0899.76104
[4] Bertozzi, A.L.; Carrillo, J.A.; Laurent, T., Blowup in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22, 683-710, (2009) · Zbl 1194.35053
[5] A.L. Bertozzi, T. Laurent, F. Léger, Aggregation via the Newtonian potential and aggregation patches, Math. Models Methods Appl. Sci. (in press). · Zbl 1241.35153
[6] Biler, P.; Karch, G.; Monneau, R., Nonlinear diffusion of dislocation density and self-similar solutions, Comm. math. phys., 294, 145-168, (2010) · Zbl 1207.82049
[7] Blanchet, A.; Calvez, V.; Carrillo, J.A., Convergence of the mass-transport steepest descent for the sub-critical patlak – keller – segel model, SIAM J. numer. anal., 46, 691-721, (2008) · Zbl 1205.65332
[8] Carrillo, J.A.; Di Francesco, M.; Figalli, A.; Laurent, T.; Slepčev, D., Global-in-time weak measure solutions, and finite-time aggregation for nonlocal interaction equations, Duke math. J., 156, 229-271, (2011) · Zbl 1215.35045
[9] Carrillo, J.A.; McCann, R.J.; Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. mat. iberoam., 19, 971-1018, (2003) · Zbl 1073.35127
[10] Carrillo, J.A.; McCann, R.J.; Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. ration. mech. anal., 179, 2, 217-263, (2006) · Zbl 1082.76105
[11] Carrillo, J.A.; Toscani, G., Asymptotic \(L^1\) decay of the porous medium equation to self-similarity, Indiana univ. math. J., 46, 113-142, (2000) · Zbl 0963.35098
[12] Castro, A.; Córdoba, D., Global existence, singularities and ill-posedness for a nonlocal flux, Adv. math., 219, 6, 1916-1936, (2008) · Zbl 1186.35002
[13] Chae, D.; Córdoba, A.; Córdoba, D.; Fontelos, M.A., Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. math., 194, 1, 203-223, (2005) · Zbl 1128.76372
[14] 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
[15] Córdoba, A.; Córdoba, D.; Fontelos, M.A., Formation of singularities for a transport equation with nonlocal velocity, Ann. of math., 162, 3, 1377-1389, (2005) · Zbl 1101.35052
[16] Córdoba, A.; Córdoba, D.; Fontelos, M.A., Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. math. pures appl. (9), 86, 6, 529-540, (2006) · Zbl 1106.35059
[17] Deslippe, J.; Tedstrom, R.; Daw, M.S.; Chrzan, D.; Neeraj, T.; Mills, M., Dynamics scaling in a simple one-dimensional model of dislocation activity, Phil. mag., 84, 2445-2454, (2004)
[18] Dong, H., Well-posedness for a transport equation with nonlocal velocity, J. funct. anal., 255, 11, 3070-3097, (2008) · Zbl 1170.35004
[19] Folland, G.B., ()
[20] Head, A.K., Dislocation group dynamics I. similarity solutions od the \(n\)-body problem, Phil. mag., 26, 43-53, (1972)
[21] Head, A.K., Dislocation group dynamics II. general solutions of the \(n\)-body problem, Phil. mag., 26, 55-63, (1972)
[22] Head, A.K., Dislocation group dynamics III. similarity solutions of the continuum approximation, Phil. mag., 26, 65-72, (1972)
[23] Hiai, F.; Petz, D., ()
[24] Jordan, R.; Kinderlehrer, D.; Otto, F., The variational formulation of the fokker – plank equation, SIAM J. math. anal., 29, 1, 1-17, (1998) · Zbl 0915.35120
[25] 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
[26] McCann, R.J., A convexity principle for interacting gases, Adv. math., 128, 1, 153-179, (1997) · Zbl 0901.49012
[27] Morlet, A.C., Further properties of a continuum of model equations with globally defined flux, J. math. anal. appl., 221, 1, 132-160, (1998) · Zbl 0916.35049
[28] Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. partial differential equations, 26, 1-2, 101-174, (2001) · Zbl 0984.35089
[29] Saff, E.B.; Totik, V., Logarithmic potentials with external fields, (1997), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0881.31001
[30] Villani, C., ()
[31] Voiculescu, D., The analogues of entropy and of fisher’s information measure in free probability theory. II, Invent. math., 118, 3, 411-440, (1994) · Zbl 0820.60001
[32] Voiculescu, D., The analogues of entropy and of fisher’s information measure in free probability theory. IV. maximum entropy and freeness, (), 293-302 · Zbl 0960.46040
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