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Gradient flow for a class of diffusion equations with Dirichlet boundary data
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In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope.
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Cited by 1 Pith paper
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Sharp Continuity Moduli for Dirichlet Heat Flow in Boundary-Reservoir Transport Metrics
The killed Dirichlet heat semigroup is Lipschitz in W_{b,1}, exactly 1/p-Hölder on mass sublevels in W_{b,p} for p>1, and discontinuous at zero, which obstructs EVI_λ realizations in W_{b,2}.
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