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Newton’s second law with a semiconvex potential

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Abstract

We make the elementary observation that the differential equation associated with Newton’s second law \(m\ddot{\gamma }(t)=-D V(\gamma (t))\) always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if \(-D V\) satisfies a one-sided Lipschitz condition. We will then build upon this idea to verify the existence of solutions for the Jeans-Vlasov equation, the pressureless Euler equations in one spatial dimension, and the equations of elastodynamics under appropriate semiconvexity assumptions.

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Acknowledgements

We acknowledge that the results verified below are not all new. The existence of weak solutions to Jeans-Vlasov equation with a semiconvex interaction potential was essentially obtained by Ambrosio and Gangbo [2]; in a recent paper which motivated this study [23], we verified the existence of solutions of the pressureless Euler system in one spatial dimension with a semiconvex interaction potential; Demoulini [11] established the existence of measure-valued solutions to the equations of elastodynamics with nonconvex stored energy; and the existence of weak solutions of the corresponding perturbed system was verified by Dolzmann and Friesecke [18]. Nevertheless, we contend that our approach to verifying existence is unifying. In particular, we present a general method to address the existence of weak and measure-valued solutions of hyperbolic evolution equations when the underlying nonlinearity is appropriately semiconvex.

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  1. Department of Mathematics, University of Pennsylvania, Philadelphia, USA

    Ryan Hynd

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  1. Ryan Hynd
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Correspondence to Ryan Hynd.

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

R. Hynd: Partially supported by NSF grant DMS-1554130.

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Hynd, R. Newton’s second law with a semiconvex potential. Partial Differ. Equ. Appl. 3, 11 (2022). https://doi.org/10.1007/s42985-021-00136-1

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