Abstract
The stability theory of entropy solutions to the compressible Euler system on a half-line with prescribed boundary conditions, such as inflow, outflow, and impermeable boundary conditions, remains largely open. It is worth emphasizing that establishing the stability of even a single reflected shock at the boundary has been a highly challenging problem. In this talk, I will briefly review the classical stability theory for small-BV entropy solutions in the whole-space setting and then turn to the half-line problem, where I will present recent progress on the stability of entropy solutions in the presence of a boundary.
Abstract: We discuss lower bounds on Lyapunov exponents for linear PDEs driven by random velocity fields, including the advection–diffusion equation and the linearized stochastic Navier–Stokes equations. In particular, we show that the exponential rate of decay of the L^2-norm for solutions to the advection–diffusion equation is optimal. A key ingredient in the proof is high-frequency stochastic instability, arising from the non-degeneracy of the driving noise. This talk is based on joint work with Martin Hairer, Tommaso Rosati, and Sam Punshon-Smith.
Abstract(임시)
: In critical PDEs, finite-time blow-up may occur while the scaling-critical norm remains bounded, with the singularity arising instead from concentration and loss of compactness. In this talk, we study this phenomenon for the radial focusing energy-critical wave equation in three dimensions. Starting from the radial free wave equation, we discuss incoming and outgoing radiation and the channel of energy method, which quantifies the energy escaping through exterior light cones. We then examine non-radiative dynamics and the rigidity principles arising from the channel of energy. Finally, we discuss how these ideas are related to the analysis of radial Type II blow-up.
[References]
1. T. Duyckaerts, C. E. Kenig, F. Merle, Universality of blow-up profile for small radial Type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. 13 (2011), 533–599.
2. C. E. Kenig, A. Lawrie, B. Liu, W. Schlag, Channels of energy for the linear radial wave equation, Adv. Math. 285 (2015).
3. T. Duyckaerts, C. E. Kenig, F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge J. Math. 1 (2013), 75–144.