  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.
