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Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1d.
| Content Provider | CiteSeerX |
|---|---|
| Author | Bonaschi, G. A. Carrillo, J. A. Francesco, M. Di Peletier, M. A. |
| Abstract | Abstract. We prove the equivalence between the notion of Wasserstein gradient flow for a onedimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L 2 gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow. 1. |
| File Format | |
| Access Restriction | Open |
| Subject Keyword | Wasserstein Gradient Flow So-called Wave-front-tracking Algorithm Regularization Effect Pseudo-inverse Distribution Function Gradient Flow Intermediate Step Repulsive Kernel Entropy Solution Scalar Conservation Law Rigorous Particle-system Approximation Onedimensional Nonlocal Transport Pde Abstract Particle Method Relies Attractive Repulsive Newtonian Potential Burgers-type Scalar Conservation Law Gradient Flow Solution |
| Content Type | Text |
| Resource Type | Article |
