Partial differential equations with quadratic nonlinearities viewed as matrix-valued optimal ballistic transport problems

DOI10.1007/S00205-022-01754-8arXiv1905.06059WikidataQ114231138 ScholiaQ114231138MaRDI QIDQ6318783

Publication date: 15 May 2019

Abstract: We study a rather general class of optimal "ballistic" transport problems for matrix-valued measures. These problems naturally arise, in the spirit of emph{Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605}, from a certain dual formulation of nonlinear evolutionary equations with a particular quadratic structure reminiscent both of the incompressible Euler equation and of the quadratic Hamilton-Jacobi equation. The examples include the ideal incompressible MHD, the template matching equation, the multidimensional Camassa-Holm (also known as the Hdiv geodesic equation), EPDiff, Euler-alpha, KdV and Zakharov-Kuznetsov equations, the equations of motion for the incompressible isotropic elastic fluid and for the damping-free Maxwell's fluid. We prove the existence of the solutions to the optimal "ballistic" transport problems. For formally conservative problems, such as the above mentioned examples, a solution to the dual problem determines a "time-noisy" version of the solution to the original problem, and the latter one may be retrieved by time-averaging. This yields the existence of a new type of absolutely continuous in time generalized solutions to the initial-value problems for the above mentioned PDE. We also establish a sharp upper bound on the optimal value of the dual problem, and explore the weak-strong uniqueness issue.

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) KdV equations (Korteweg-de Vries equations) (35Q53) Variational methods applied to PDEs (35A15) Viscoelastic fluids (76A10) Variational problems in a geometric measure-theoretic setting (49Q20) Magnetohydrodynamics and electrohydrodynamics (76W05) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Weak solutions to PDEs (35D30) Existence, uniqueness, and regularity theory for incompressible inviscid fluids (76B03) Euler equations (35Q31) Optimal transportation (49Q22)

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