Some relatively new techniques for nonlinear problems.
Some remarks on Prandtl system
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.
Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations
This article addresses some theoretical questions related to the choice of boundary conditions, which are essential for modelling and numerical computing in mathematical fluids mechanics. Unlike the standard choice of the well known non slip boundary conditions, we emphasize three selected sets of slip conditions, and particularly stress on the interaction between the appropriate functional setting and the status of these conditions.
Some remarks on the KP system of the Camassa-Holm hierarchy.
Some remarks on the Navier-Stokes equations with regularity in one direction
We review the developments of the regularity criteria for the Navier-Stokes equations, and make some further improvements.
Some remarks on the non-uniqueness of the stationary solutions of Navier-Stokes equations.
Some remarks to the compactness of steady compressible isentropic Navier-Stokes equations via the decomposition method
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...
Some results on invariant measures in hydrodynamics
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 results on the asymptotic behaviour of Hopf weak solutions to the Navier-Stokes equations in unbounded domains.
Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence
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...
Some rigorous results on the Pauli-Fierz model of classical electrodynamics
Some stability results for reactive Navier-Stokes-Poisson systems
We review the main results concerning the global existence and the stability of solutions for some models of viscous compressible self-gravitating fluids used in classical astrophysics.
Some theoretical results concerning non newtonian fluids of the Oldroyd kind
Some time-dependent Hartree equations
Space of local fields in integrable field theory and deformed abelian differentials.
Space-time variational saddle point formulations of Stokes and Navier–Stokes equations
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...
Sparse grids for the Schrödinger equation
We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore...
Spatial analyticity of solutions of a nonlocal perturbation of the KdV equation.
Spatial decay estimates for the Forchheimer fluid equations in a semi-infinite cylinder
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.
Spatial patterns for the three species Gross-Pitaevskii system in the plane.