In this talk, we investigate the stability of stationary solutions to scalar viscous　conservation laws with delay effects. To establish the stability of equilibrium states, we need to impose certain conditions that depend on both the delay parameter and the initial history. This situation differs significantly from the case of the standard scalar viscous conservation law without delay. In this setting, we further investigate the stability of stationary solutions to scalar viscous conservation laws with delay effects, with the aim of deriving stability conditions that explicitly account for the effects of delay.

We consider some shallow water models that admit various steady solutions depending on the prescribed boundary conditions and bottom topography. In particular, discontinuous steady solutions arise when the flow undergoes a transition from a supercritical to a subcritical regime, whereas continuous transcritical solutions occur in transitions from subcritical to supercritical flow. We investigate the linear stability of these steady flows. This talk is based on the ongoing project with Miguel Rodrigues (University of Rennes) and Nina Aguillon (Sorbonne University).

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.

The center manifold theorem provides a framework for studying local dynamics near an equilibrium on a lower-dimensional invariant manifold. In this talk, we introduce the theorem and apply it to establish the existence of small-amplitude viscous shock profiles for the one-dimensional compressible Navier–Stokes–Fourier (NSF) system. This system describes the motion of a compressible viscous fluid with heat conduction. For an ideal gas, we restrict the profile equations to a suitable invariant curve and obtain a scalar ordinary differential equation. We then construct a profile connecting two nearby states satisfying the Rankine–Hugoniot and Lax entropy condition. References: [1] A. Bressan, A Tutorial on the Center Manifold Theorem, lecture notes. [2] A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws, Journal of Differential Equations 56 (1985), 229–262.

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.

Singularly perturbed differential equations arise in many problems in fluid mechanics and related areas, where small diffusivity or viscosity can generate sharp boundary, interior, and corner layers. These multiscale structures present significant challenges for both classical numerical methods and standard neural network approaches. We develop neural network methods that use asymptotic analysis to identify and incorporate the dominant structure of singular layers directly into the approximation. By explicitly incorporating the leading singular components into the neural network ansatz, the trainable component is used primarily to approximate the smoother part of the solution. We also consider conservative formulations based on finite volume residuals, which incorporate local flux balance into the learning framework. Numerical examples for a range of singularly perturbed problems demonstrate the accuracy and robustness of these approaches as the perturbation parameter tends to zero.