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Constructing a relativistic heat flow by transport time steps. (English) Zbl 1181.80001
The purpose of the paper is to build weak solutions of some relativistic heat equation using some time discretization scheme which involves an optimal transportation problem. The starting equation is $$\partial _{t}\rho =\text{ div}(\rho \nabla \rho /\sqrt{\rho ^{2}+|\nabla \rho |^{2}})$$ but it can be generalized as $$\partial _{t}\rho =\text{div}(\rho \nabla c^{\ast }(\nabla F^{\prime }(\rho )))$$ where $$c^{\ast }$$ is a convex function on $$\mathbb{R} ^{d}$$ and $$F$$ is a convex function on $$[0,\infty [$$. Here $$c^{\ast }$$ is the Legendre transform of some cost function $$c$$ and $$F$$ represents the entropy. In the case of the starting equation, this cost function $$c$$ is hemispherical and discontinuous. As observed in the introduction of the paper, this kind of equation covers a large number of known equations. The main tool of the paper consists to introduce some optimal transportation strategy. $$\Omega$$ being some smooth, bounded and convex domain of $$\mathbb{R}^{d}$$, let $$P(\Omega )$$ be the set of Borel probability measures on $$\Omega$$ and $$h$$ be the time step. For a given $$\rho _{0}\in P(\Omega )$$, the optimal transportation problem consists to find $$\rho ^{h}(t,y)\in P([0,T]\times \Omega )$$ through $$\rho ^{h}(0,y)=\rho _{0}(y)$$ and $$\rho ^{h}(t,y)=\rho _{i}^{h}(y)$$ for $$t\in ]ih;(i+1)h]$$ where $$\rho _{i}^{h}$$ is the solution of some minimization problem associated to a functional $$I$$ which involves the entropy $$F$$ and the cost function $$c$$. The main result of the paper establishes that the limit $$\overline{\rho }$$ of $$\rho ^{h}$$ satisfies a relativistic heat equation in the sense of distributions, assuming that $$\rho _{0}$$ is bounded from below and from above and assuming further hypotheses on $$c$$ and $$F$$. For the proof of this convergence result, the authors use the BV function framework. They first prove an existence and uniqueness result for the minimization problem associated to the optimal transportation problem. They also study a mollifier approximation of this minimization problem.

##### MSC:
 80A10 Classical and relativistic thermodynamics 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 80M50 Optimization problems in thermodynamics and heat transfer
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