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.
In critical PDEs, finite-time blow-up may occur while the scaling-critical norm remains bounded, with singularity formation arising from concentration and loss of compactness. In this first talk, we consider the radial focusing energy-critical wave equation in three dimensions and discuss the role of concentration-compactness and profile decomposition in the study of Type II blow-up. We then introduce radiation for the radial free wave equation and the channel of energy method. These ideas will provide the starting point for the rigidity analysis in the second talk.
[References]
1. H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175.
2. C. E. Kenig, A. Lawrie, B. Liu, W. Schlag, Channels of energy for the linear radial wave equation, Adv. Math. 285 (2015), 877–936.
In this second talk, we apply the channel of energy method to the nonlinear analysis of radial Type II blow-up for the three-dimensional focusing energy-critical wave equation. We discuss how radiation estimates interact with concentration and compactness to produce rigidity, and how this leads to the classification of possible blow-up profiles. Finally, we relate these ideas to the soliton resolution picture for bounded radial solutions.
[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. 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.