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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.

MSC:
35Q35 PDEs in connection with fluid mechanics
35C06 Self-similar solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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