We consider here elliptical systems as Stokes and Navier-Stokes problems in a bounded domain, eventually multiply connected, whose boundary consists of multi- connected components. We investigate the solvability in L^p theory, with 1 < p < \infty, under non standard boundary conditions. We consider also the case of Navier boundary conditions. The main ingredients for this solvability are given by the Inf-Sup conditions, some Sobolev’s inequalities for vector fields and the theory of vector potentials. These inequalities play a fundamental key and are obtained thanks to Calderon-Zygmund inequalities and integral representations. In the study of ellpitical problems, we consider both generalized solutions and strong solutions that very weak solutions. In a second part, we will consider the nonstationary case for the Stokes equations.

Tertiary oil recovery is the process which consists in injecting a solvent through a well in an underground oil reservoir, that will mix with the oil and reduce its viscosity, thus enabling it to flow towards a second reservoir. Mathematically, this process is represented by a coupled system of an elliptic equation (for the pressure) and a parabolic equation (for the concentration). The parabolic equation is strongly convection-dominated, and discretising the convection term properly is therefore essential to obtain accurate numerical representations of the solution. One of the possible discretisation techniques for this term involves using characteristic methods, applied on the test functions. This is called the Eulerian-Lagrangian Localised Adjoint Method (ELLAM). In practice, due to the ground heterogeneities, the available grids can be non-conforming and have cells of various geometries, including generic polytopal cells. Along with the non-linear and heterogeneous/anisotropic diffusion tensors present in the model, this creates issues in the discretisation of the diffusion terms. In this talk, we will present a generic framework, agnostic to the specific discretisation of the diffusion terms, to design and analyse ELLAM schemes. Our convergence result applies to a range of possible schemes for the diffusion terms, such as finite elements, finite volumes, discontinuous Galerkin, etc. Numerical results will be presented on various grid geometries.

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigenfunctions in the high energy (semiclassical) limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions (joint work with Bogdan Georgiev).

I shall discuss on control problems governed by the partial differential equations-mainly compressible Navier-Stokes equations, visco-elastic flows. I shall mention some of the basic tools applicable to study the control problems. We mainly use spectral characterization of the operator associated to the linearized PDE and Fourier series techniques to prove controllability and stabilizability results. I shall also indicate how the hyperbolic and parabolic nature of equations affects their main controllability results. Then some of our recent results obtained in this direction will be discussed.

We consider a finite volume element method for approximating the solution of a time fractional diffusion problem involving a Riemann-Liouville time fractional derivative of order alpha between 0 and 1. For the spatially semidiscrete problem, we establish optimal with respect to the data regularity L2(X)-norm error estimates, for the cases of smooth and middly smooth initial data, i.e., v in H2(X) intersection H^1_0(X) and v in H10(X). For non-smooth data v in L2(X), the optimal L2(X)-norm estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a con- sequence a quasi-optimal error estimate is established in the L^infty(X)-norm. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are developed, and error estimates are derived for both smooth and nonsmooth initial data. Finally, some numerical results are presented to illustrate the theoretical results.

My emphasis in this talk will be on functional analytical tools to the solvability and uniqueness of solutions to the nonhomogeneous boundary value problems, dealing with degenerate PDEs of elliptic type. My aim is to consider possibly general class of weights. In particular, I consider the $B_{p}$-class of weights, introduced by Kufner and Opic, which is much more general class than the commonly studied Muckenhoupt $A_{p}$-class.

We consider the one dimensional compressible Navier-Stokes system near a constant steady state with the periodic boundary conditions. The linearized system around the constant steady state is a hyperbolic-parabolic coupled system. We discuss some of the properties of the linearized system and its spectrum. Next we study some controllability results of the system.