This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.
We study the nonstationary Navier-Stokes equations in the entire three-dimensional space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.
Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the -solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form. In the second kind decomposition the -solenoidal...
This is a survey of some recent results on the existence of globally defined weak solutions to the Navier-Stokes equations of a viscous compressible fluid with a general barotropic pressure-density relation.
We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.
The purpose of this paper is to correct some drawbacks in the proof of the well-known Boundary Layer Theory in Oleinik’s book. The Prandtl system for a nonstationary layer arising in an axially symmetric incopressible flow past a solid body is analyzed.
We review the developments of the regularity criteria for the Navier-Stokes equations, and make some further improvements.
In [18]–[19], P.L. Lions studied (among others) the compactness and regularity of weak solutions to steady compressible Navier-Stokes equations in the isentropic regime with arbitrary large external data, in particular, in bounded domains. Here we investigate the same problem, combining his ideas with the method of decomposition proposed by Padula and myself in [29]. We find the compactness of the incompressible part of the velocity field and we give a new proof of the compactness of the “effective...
In questa nota, si presentano risultati di esistenza e di unicità di misure invarianti per l'equazione di Navier-Stokes che governa il moto di un fluido viscoso incomprimibile omogeneo in un dominio bidimensionale soggetto a una forzante che ha due componenti: una deterministica e una di tipo rumore bianco nella variabile temporale.
Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular...
The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities u and pressure p. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between H1 and H'2, both Hilbert spaces H1 and H2 being Cartesian products of (intersections of) Bochner spaces, or duals of those. Based on these results, the operator that corresponds to the Navier−Stokes...
The spatial behavior of solutions is studied in the model of Forchheimer equations. Using the energy estimate method and the differential inequality technology, exponential decay bounds for solutions are derived. To make the decay bounds explicit, we obtain the upper bound for the total energy. We also extend the study of spatial behavior of Forchheimer porous material in a saturated porous medium.