Partial Differential Equations with Quadratic Nonlinearities Viewed as Matrix-Valued Optimal Ballistic Transport Problems

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