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 Brenier (Commun Math Phys 364(2):579–605, 2018), 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 $$H({{\,\mathrm{div}\,}})$$ 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.