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

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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.

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Notes

  1. This definition of subsolutions does not take into account the corresponding wave cone that plays a significant role in the theory of convex integration, cf. [20, 21]. However, in the case of the incompressible Euler equation this definition is known to be equivalent [19] to the conventional definition of subsolutions. When the wave cone is not very large, cf. [26, 68], there is a discrepancy between the two definitions.

  2. In the case \(n=1\) this definition would imply that \(v=0\) almost everywhere in the set \([I + 2B=0]\). The generalized solution to the inviscid Burgers equation provided by [11, Proposition 4.1] coincides with (4.8) in the support of the measure \(I\mu \otimes \hbox {d}t+2B\). However, generally speaking, it does not vanish outside of the support. This is easy to check by considering the initial datum \(v_0(x)=sign \left( x-\frac{1}{2}\right) ,\ x\in [0,1]\simeq {\mathbb {T}}^1\) for which the solution of [11, Proposition 4.1] can be computed by hand.

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Acknowledgements

The author wishes to thank the anonymous referees for many valuable comments, which led to important improvements of the manuscript. The author is very grateful to Yann Brenier and Wenhui Shi for inspiring discussions on the subject. The research was partially supported by the Portuguese Government through FCT/MCTES and by the ERDF through PT2020 (projects UID/MAT/00324/2020,

PTDC/MAT-PUR/28686/2017 and TUBITAK/0005/2014).

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    Dmitry Vorotnikov

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Vorotnikov, D. Partial Differential Equations with Quadratic Nonlinearities Viewed as Matrix-Valued Optimal Ballistic Transport Problems. Arch Rational Mech Anal 243, 1653–1698 (2022). https://doi.org/10.1007/s00205-022-01754-8

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