Partial Differential Equations and their Applications Seminar
The seminar meets on Tuesdays at noon, in B3.02.
Term 1 - 26-27
No seminar
Title: Hamilton-Jacobi equations with partial coercivity: regularity via controllable dynamics
Abstract:
Hamilton-Jacobi equations are a key tool in optimal control theory and play a central role in mean-field games. Although first-order equations lack parabolic smoothing, viscosity solutions are nevertheless H枚lder continuous provided that the Hamiltonian grows superlinearly in the momentum variable. However, this coercivity assumption fails whenever the dynamics do not allow control in all directions. Partially coercive Hamilton-Jacobi equations thus arise naturally in the optimal control of systems with constrained or degenerate dynamics, including a broad class of linear control systems.
Even when direct control is unavailable in some directions, the dynamics may still be controllable if the control interacts with the drift in a suitable way. For linear systems this is characterised by the Kalman rank condition. I will present recent work on a class of first-order Hamilton-Jacobi equations motivated by such controllable linear systems. We show that, under the Kalman rank condition, viscosity solutions are H枚lder continuous for Hamiltonians with superlinear growth in the controlled directions only. The resulting H枚lder regularity is anisotropic, reflecting the geometry induced by the underlying control structure. This is joint work with Alp谩r M茅sz谩ros (Durham).
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Abstract:
Title: Self-similarity and diffusive limits for linear kinetic equations
Abstract:
In this talk, I will present recent results obtained with Jos茅 Ca帽izo and St茅phane Mischler regarding the asymptotic behaviour of some linear kinetic equations without confinement. We present the results for three types of operators and their fractional variants.The unconfined setting prevents the use of standard hypocoercivity arguments and, to the best of our knowledge, the decay of the L2norm of the solution in an unconfined setting was only proved recently. We improve these results by providing a more detaileddescription of the asymptotics, not only establishing norm decay but also identifying the exact asymptotic profile of the solution.After rescaling, this allows us to show a mesoscopic-to-macroscopic limit uniformly in time, which is also new. At the end of the talk,I will briefly present ongoing extensions of these results to more general collision operators.
