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

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.