The 3D-Navier Stokes equations can be stabilized by tangential boundary controller that moreover are finite dimensional. We also present stabilization results for the Boussinesq system. This is joint work with I. Lasiecka and B. Priysad.

We consider a system consisting of a nonlinear advection-diffusion equation coupled with an elastically supported Mindlin-type plate equation with time as a parameter. This type of a system arises in the context of geomorphology and geodynamics applications to landscape evolution modeling. The advection-diffusion equation describes changes in landscape surface elevation in response to fluvial and hillslope bedrock erosion and uplift relative to magma emplacement. The proposed elastically supported Mindlin-type plate equation is used to capture uplift-rate deformation associated with magmatic intrusions. We establish the existence, uniqueness, and continuous dependence on data of a global weak solution to the coupled system. The proof is based on the Faedo-Galekin technique with a compactness argument, energy estimates derived by means of a Gronwall-type argument, and the energy method combined with the penalty function technique. Using the semigroup approach, we also obtain a unique local strong solution to the initial-boundary value problem for the advection-diffusion equation given surface uplift as data. This is joint work with Luther W. White, Oregon Applied Mathematics Institute.

The initial value problem for nonlinear evolution equations has been studied extensively and from many points of view over the last fifty years. To the contrary, initial-boundary value problems for these equations remain widely unexplored despite the fact that such problems arise naturally in applications. This talk is concerned with a new approach for the well-posedness of nonlinear initial-boundary value problems, which combines the linear solution formulae produced via the unified transform method of Fokas with suitably adapted harmonic analysis techniques. Concrete examples to be discussed include the nonlinear Schrödinger and Korteweg-de Vries equations, as well as a reaction-diffusion equation with power nonlinearity, formulated either on the half-line or on a finite interval.

The Kuramoto-Sivashinsky equation (KSE) is a highly chaotic dynamical system that arises in flame fronts, plasmas, crytral growth, and many other phenomena. Due to its lack of a maximum principle, the KSE is often studied as an analogue to the 3D Navier-Stokes equations (NSE). Much progress has been made on the 1D KSE since roughly 1984, but for the 2D KSE, even global well-posedness remains a major open question. In analogy with regularizations of the 3D NSE, we present modifications of the 2D KSE which allow for global well-posedness, while still retaiing many imporant features of the 2D KSE. However, as has been demonstrated recently by Kostianko, Titi, and Zelik, standard regularizations, that work well for Navier-Stokes, fail when applied to even the 1D KSE. Thus, we present entirely new types of modification for the 2D KSE. This talk will describe key ideas of the analysis, and also show many colorful movies of solutions.

Flutter is a self-excitation instability of an elastic structure surrounding fluid flow. Motivated by piezoelectric energy harvesting, we consider the large deflections of an inextensible cantilever. Mathematically, there is minimum rigorous analysis of this recent PDE model. For the large deflections of the cantilever, nonlinear restoring forces are present due to the inextensibility constraint (rather than local stretching). This results both nonlinear stiffness and inertia terms which lead to quasilinear and nonlocal equations. To obtain existence of solutions, we utilize the Galerkin procedure with cantilever modes. Due to the nonlinearity, there is no natural weak formulation, and identifying weak limits requires additional compactness, forcing higher topologies for smooth data. The addition of strong (Kelvin-Voigt) structural damping is necessary in order to obtain any a-priori estimates when nonlocal inertia is included. Local existence of strong solutions for the full system is then obtained. Uniqueness follows from a novel decomposition of the dynamics.

In accurately simulating turbulent flows, two major difficulties arise before the simulation begins; namely the problem of determining the initial state of the flow, and the problem of estimating the parameters. Data assimilation helps to resolve the first problem by eliminating the need for complete knowledge of the initial state. It incorporates incoming data into the equations, driving the simulation to the correct solution. Recently, a promising new data assimilation algorithm (the AOT algorithm) has been proposed by Azouani, Olson, and Titi, which uses a feedback control term to incorporate observations at the PDE level. This talk is focused on using the AOT algorithm in the context of measurements devices which move in time, such as satellites or drones. We find that, in the context of the Allen Cahn equation, by moving the sampling points dynamically, we can greatly reduce the number of sampling points required, while achieving better accuracy.