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.
Abstract: A central challenge in constructing entropy solutions of hyperbolic conservation laws is passing to the limit in nonlinear approximation schemes. Uniform (L^\infty) bounds provide weak compactness, but this alone is insufficient to identify the limit of the nonlinear flux. This talk introduces compensated compactness as a method for overcoming this difficulty by exploiting the entropy structure of the equations. We explain how compactness of entropy production yields constraints on oscillations in approximate solutions and how, under suitable structural assumptions, these constraints lead to strong convergence. We then discuss how this approach can be used to establish the existence of entropy solutions through vanishing viscosity and relaxation approximations.
References
[1] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer, 2016.
[2] R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), 383–420.
Abstract: Fenichel’s first theorem provides a framework for reducing singularly perturbed ordinary differential equations to dynamics on invariant manifolds. In this talk, we introduce the theorem and explain its application to the construction of small-amplitude viscous shock profiles for the Brenner–Navier–Stokes–Fourier (BNSF) system. We first discuss critical manifolds, normal hyperbolicity, and local invariance. We then transform the traveling-wave equations into a slow–fast system and verify the hypotheses of Fenichel’s theorem. The resulting invariant curve reduces the profile problem to a scalar ordinary differential equation, from which a connection between the prescribed end states is constructed.
Finally, we outline the derivative estimates used to establish monotonicity of the viscous shock profiles.
Reference:
[1] S. Eo, N. Eun, M.-J. Kang, and H. Oh, Traveling wave solutions to Brenner–Navier–Stokes–Fourier system, Journal of Differential Equations 422 (2025), 639–658.
[2] C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems, R. Johnson (ed.), Lecture Notes in Mathematics, vol. 1609, Springer, pp. 44–118, 1995.
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.