We study the two-dimensional acoustical rarefaction waves under the irrotational assumptions. We provide a new energy estimates without loss of derivatives. We also give a detailed geometric description of the rarefaction wave fronts. As an application, we show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves. This is a joint work with Prof. Pin Yu in Tsinghua University.

In this talk we prove the existence of solutions to an optimal control problem where the constraint is an ill-posed, nonlinear equation containing a Fractional Laplacian. For any fixed control data, the constraint equation is known to have multiple solutions by a previous application of the Mountain Pass Theorem. Due to the pointwise nature of the conditions on controls needed to invoke this theorem, we must make substantial adaptations to the usual direct method of calculus of variations in order to prove our main existence result. The main theoretical tools are a thoughtful construction of an admissible set of controls, and a technical lemma that ensures that a minimizing sequence of pairs of controls and corresponding states exhibits convergence to another control-state pair that satisfies the constraint equation.

In this talk, we will introduce the result of the nonlinear stability threshold for compressible Couette flow, highlighting several key innovations. First, we introduce a new quantity that significantly weakens the lift-up effect, which is the key difficulty in shear flow problems. Second, we utilize the properties of acoustic waves to achieve crucial cancellations, a method that fundamentally differs from the incompressible case. Third, we propose a new set of decoupled diffusion waves, improving the decay of errors. This approach contrasts with previous constructions of coupled diffusion waves and can be extended to more general hyperbolic-parabolic coupled systems. Additionally, we employ a Poincaré-type inequality, aided by Huang-Li-Matsumura’s inequality, which plays an important role in managing certain critical (for time) energy estimates.

There is a rich history of the study of traveling wave solutions to the free boundary Euler equations - otherwise known as the water wave problem; nevertheless, it is only within the last six years that the analogous study of traveling wave solutions to free boundary fluids with viscosity - such as the incompressible Navier-Stokes equations - has commenced. Even more recently, this traveling wave study has expanded to a larger family of dissipative fluid systems including compressible flows, fluids obeying Darcy’s law, and the shallow water equations. This talk focuses specifically on the shallow water equation’s version of the water wave problem. The shallow water equations, which are derived from the free boundary Navier-Stokes equations with Navier slip boundary conditions via a rescaling, asymptotic expansion, and depth-averaging procedure, are both mathematically and computationally important. For these equations we shall discuss three recent results: (1) the traveling wave problem and the limits of vanishing viscosity and capillarity, (2) the existence of families of two-dimensional roll wave solutions, and (3) the stationary wave problem with variable bathymetry and the nature of large solutions. These theorems are unified by the fact the main engine of their proofs is the implicit function theorem, although each result utilizes a different manifestation. These are a Nash-Moser variant to handle derivative loss, a multiparameter bifurcation theorem, and an analytic global implicit function theorem.

In this talk, we present an exponential-integrator based mulitiindex Monte Carlo method (MIMC) for weak approximations of mild solutions of semilinear stochastic partial differential equations (SPDE). We present recent theoretical results on multiindex coupled solutions of the SPDE are presented, namely that such couplings are stable and satisfy multiplicative error estimates, and describe how this theory can be utilized to obtain a tractable MIMC method. Numerical examples show that MIMC outperforms alternative methods, such as multilevel Monte Carlo, in settings with low regularity.