Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE

By Lawrence C. Evans. 2010

108 citations recorded by the source corpus.

We investigate the vanishing viscosity limit for Hamilton–Jacobi PDE with nonconvex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is to introduce a solution σε of the adjoint of the formal linearization, and then to integrate by parts with respect to the density σε. This procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2ef431309ee43e19e33d72b76a1fddbc35f7eb11

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Adjoint methods for static Hamilton–Jacobi equations (2009, methodology): “The adjoint equation here is very nice and somehow similar to the ne that Evans introduce in [7] of the time-dependent case.”
On the existence of classical solutions for stationary extended mean field games (2013, background): “Evans ([Eva10]) to prove W 1,∞ bounds for u for a broad class of Hamiltonians.”
Time dependent mean-field games in the superquadratic case (2013, methodology): “The key tools are the adjoint method [16] and the methods developed in [22] (see also [29]).”
Nonlinear elliptic systems and mean-field games (2016, methodology): “1 we show a more precise connection among the classical systems of [14, 15] and systems of the form (1), which is related to the adjoint methods for Hamilton-Jacobi equation explored by Evans [30].”
Selection problems for a discount degenerate viscous Hamilton–Jacobi equation (2017, unknown): “See [12, 32, 7, 6] for instance.”
Time-Dependent Mean-Field Games in the Subquadratic Case (2013, methodology): “To do so we use the nonlinear adjoint method introduced in [17], see also the applications in [6–8, 18, 51].”
A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians (2012, unknown): “A version of the second term in (i) was first derived by Evans [8].”