Newton’s second law with a semiconvex potential

Ryan Hynd ()
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Ryan Hynd: University of Pennsylvania

Partial Differential Equations and Applications, 2022, vol. 3, issue 1, 1-34

Abstract: Abstract We make the elementary observation that the differential equation associated with Newton’s second law $$m\ddot{\gamma }(t)=-D V(\gamma (t))$$ m γ ¨ ( t ) = - D V ( γ ( t ) ) always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if $$-D V$$ - 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.

Keywords: 35L65; 60B10; 26B25; 35D30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-021-00136-1

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