Penel Patrick

Université de Toulon | USTV · Institute of ingineering sciences of Toulon

We deal with a weak solution \({{\textbf {v}}}\) to the Navier–Stokes initial value problem in \({{\mathbb {R}}}^3\times (0,T)\), that satisfies the strong energy inequality. We impose conditions on certain spectral projections of \({\varvec{\omega }}:={\textbf {curl}}\, {{\textbf {v}}}\) or just \({{\textbf {v}}}\), and we prove the regularity of...

The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1 , (ζ2)+ (the positive part of ζ2 ), and ζ3 , where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor...

We assume that \(\Omega ^{t}\) (for t ∈ [0, T]) is a time-varying domain in \(\mathbb{R}^{3}\). Particularly, \(\Omega ^{t}\) can be a region around colliding bodies. Under certain conditions on \(\Omega ^{t}\) and the way it varies, we prove the weak solvability of the Navier–Stokes system with Navier’s slip boundary condition in \(Q_{(0,T)}:=\{ (...

We consider the incompressible Navier-Stokes equations in the entire three-dimensional space. Assuming additional regularity on the components of the vector field ∂ 3 u we show intermediate anisotropic regularity results between the results by I. Kukavica and M. Ziane [J. Math. Phys. 48, No. 6, 065203, 10 p. (2007; Zbl 1144.81373)] and by C. Cao an...

We show that a smooth solution u 0 of the Euler boundary value problem on a time interval (0, T 0) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T 0). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “natu...

We deal with a weak solution v to the Navier-Stokes initial value problem in R^3 x(0,T). We denote by \omega^+ a spectral projection of \omega=\curl\, v, defined by means of the spectral resolution of identity associated with the self-adjoint operator \curl. We show that certain conditions imposed on \omega^+ or, alternatively, only on \omega^+_3 (...

We denote P-a(+) := integral(infinity)(a) dE(lambda), where {E-lambda} is the spectral resolution of identity associated with the self-adjoint operator curl in the space L-sigma(2) (R-3). Further, we denote omega(+)(a) := P-a(+) curl v, where v is a weak solution to the Navier-Stokes initial value problem in R-3 x (0, T). We assume that a = a(t) is...

We prove the existence of a steady solution to the Navier–Stokes equations for barotropic compressible fluid in a bounded simply connected domain with the prescribed generalized impermeability conditions u · n = 0, curl u · n = 0 and curlu · n = 0 on the boundary, we assume that the state law for the pressure has the form P(ρ) = ρ for . We prove se...

We prove the existence of a partially strong solution to the steady Navier–Stokes equations for barotropic compressible fluid in a bounded simply connected domain with the prescribed generalized impermeability conditions u n = 0, curl u n = 0 and curl2u n = 0 on the boundary. We assume that the state law for the pressure has the form for γ > 3. We...

There exists a series of other works dealing with flows in time varying domains that concern the motion of one or more bodies in a fluid. The fluid and the bodies are studied as an interconnected system so that the position of the bodies in the fluid is not apriori known. The weak solvability of such a problem, provided the bodies do not touch each...

In this Note, we prove the existence of strong solutions to the Navier–Stokes equations for incompressible viscous fluids in a general regular bounded domain of R3 on a “short” time interval (0,T0), independent of the viscosity and of the friction between the fluid and the boundary. The solutions to the Navier–Stokes problem satisfy the inhomogeneo...

In this short note we consider the 3D Navier–Stokes equations in the whole space, for an incompressible fluid. We provide sufficient conditions for the regularity of strong solutions in terms of certain components of the velocity gradient. Based on the recent results from Kukavica (J Math Phys 48(6):065203, 2007) we show these conditions as anisotr...

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists To > 0, ν* > 0 and a unique family of strong solutions uν of the Euler or Navier-Stokes initial-boundary value problem on the time interval (0, T0), depending continuously on the viscosity coefficient v for 0 < ν < ν*. The solutions of th...

In this Note, we prove the existence of a partially strong solution to the steady Navier–Stokes equations for viscous barotropic compressible fluids, in a bounded simply connected domain of R3 with the prescribed generalized impermeability conditions curlku⋅n=0, k=0,1,2 on the boundary. We call the solution “partially strong” because only the diver...

The Navier–Stokes equations for a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H1 as well as the mass force such that the stationary density is uniquely determined but admits vacua. Missing uniform lower bound for the density is compensated by a careful modific...

We study the Oseen problem with rotational effect in exterior three-dimensional domains. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces in the whole three-dimensional space. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type and the theory of Caldero...

Provided the initial velocity and the external body force are sufficiently smooth, there exist T(0) > 0, nu* > 0 and a unique continuous family of strong solutions u(v) (0 <= nu <= nu*) of the Euler or Navier-Stokes initial-boundary value problem on the time interval (0, T(0)). The solutions of the Navier-Stokes problem satisfy a Navier-type bounda...

In this Note, we treat the Navier–Stokes equation with slip Navier's boundary condition in a time variable domain around a finite system of compact bodies moving in a container. The motion of the bodies is assumed to be a priori known. The bodies may collide at a finite number of time instants. We present the theorem on the global in time existence...

We assume that t (fort2 (0;T )) is a time varying domain inR 3 , which is the exterior of several compact bodies moving in a container and striking at time instants t2T c , whereT c is a finite subset of (0;T ). We consider the Navier-Stokes equation with Navier's slip boundary condition and we prove its weak solvability in Q(0;T ) :=f(x;t); 0 < t...

We consider time-periodic Oseen flow around a rotating body in ℝ 3 . We prove a priori estimates in L q -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates, the problem is reduced to a stationary Oseen equation with the additional t...

In this Note, we establish new estimates for the long time behavior of the solutions to the Navier–Stokes Equations for a compressible barotropic fluid in 1D, with homogeneous Dirichlet boundary conditions, with large initial data, and under the influence of a large mass force in the case when the stationary density admits vacua: a highly singular...

We show that if (v; p) is a suitable weak solution to the Navier-Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg - see [1]) such that v3 (the third component of v) is essentially bounded in a subdomain D of a time-space cylinder QT then v has no singular points in D.

The global behavior of the solutions to one-dimensional Navier-Stokes system with a free boundary is investigated for large data. It is shown that the solutions stabilize to equilibrium in general on subsequences, and completely, if the body force is such that the corresponding equilibrium is unique. Mild condition on state equation is imposed whic...

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the bounda...

We deal with the Navier–Stokes equation with the generalized impermeability bound-ary conditions. We give basic information on these boundary conditions and we further study dynamical properties of solutions, like stability, fast decay and similar. The used norms are graph norms of powers of the Stokes operator S.

We study the asymptotic behaviour in time of incompressible non-Newtonian fluids in the whole space assuming that initial data also belong to L1. Firstly, we consider the weak solution to the power-law model with non-zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with...

We formulate sufficient conditions for regularity up to the boundary of a weak solution v in a subdomain Ω×(t 1 ,t 2 ) of the time-space cylinder Ω×(0,T) by means of requirements on one of the eigenvalues of the rate of deformation tensor. We assume that Ω is a cube.

We consider a generic scalar model for the Oseen equations in an exterior three-dimensional domain. We assume the case of a non-constant coefficient function. Using a variational approach we prove new regularity properties of a weak solution whose existence and uniqueness in anisotropically weighted Sobolev spaces were proved in [S. Kračmar and P....

We extend sufficient conditions for regularity we described in our previous works so that they are valid not only in the interior, but up to the boundary of a flow field. The conditions are based on the integrability properties of either one of the eigenvalues of the rate of deformation tensor or one component of velocity.

We treat the Stokes and the Navier-Stokes equation with the conditions curlku · n = 0 (k = 0, 1, 2) on the boundary of the flow field. The approach is based on a spectral analysis and properties of operator curl. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

In this Note, we formulate sufficient conditions for regularity of a so called suitable weak solution (v;p) in a sub-domain D of the time–space cylinder QT by means of requirements on one of the eigenvalues or on the eigenvectors of the symmetrized gradient of velocity. To cite this article: J. Neustupa, P. Penel, C. R. Acad. Sci. Paris, Ser. I 336...

We formulate sufficient conditions for regularity of a suitable weak solution(v; p ) in a sub—domainD of the time—space cylinderQT in Section 3. The conditions are anisotropic in the sense that the assumptions about vi v2 (the first two components of velocity) differ from the assumptions about the third component of velocity v3. The question what t...

Mathematical modeling and numerical simulation in fluid mechanics are topics of great importance both in theory and technical applications. The present book attempts to describe the current status in various areas of research. The 10 chapters, mostly survey articles, are written by internationally renowned specialists and offer a range of approache...

We study the asymptotic behaviour of the certain class of non-Newtonian incompressible fluids-power law fluids in the whole space when the external force is zero. Assuming that initial data belong toL 1∩L 2 we prove thatL 2 decay in time ist −1/4.

We formulate su cient conditions for regularity of a so-called suitable weak solution (v; p) in a sub-domain D of the time-space cylinder Q T by means of requirements on one of the eigenvalues or on the eigenvectors of the rate of deformation tensor. KeywordsNavier-Stokes equations