共查询到20条相似文献,搜索用时 15 毫秒
1.
We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982). 相似文献
2.
H. Beirão da Veiga 《Journal of Mathematical Fluid Mechanics》2009,11(2):233-257
In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear
dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the
second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space
\mathbbR+n{\mathbb{R}}_+^n under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs.
We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume
periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain
Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary
to the evolution problem. 相似文献
3.
H. Beirão da Veiga 《Journal of Mathematical Fluid Mechanics》2009,11(2):258-273
In this paper we consider a class of stationary Navier–Stokes equations with shear dependent viscosity, in the shear thinning
case p < 2, under a non-slip boundary condition. We are interested in global (i.e., up to the boundary) regularity results, in dimension n = 3, for the second order derivatives of the velocity and the first order derivatives of the pressure. As far as we know,
there are no previous global regularity results for the second order derivatives of the solution to the above boundary value
problem.
We consider a cubic domain and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume
periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω
and simultaneously with a flat boundary. The extension to non-flat boundaries is done in the forthcoming paper [7], by following
ideas introduced by the author, for the case p > 2, in reference [5]. The results also hold in the presence of the classical convective term, provided that p is sufficiently close to the value 2.
相似文献
4.
In this paper, we investigate the vanishing viscosity limit for solutions to the Navier–Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R 3. We first obtain the higher order regularity estimates for the solutions to Prandtl’s equation boundary layers. Furthermore, we prove that the strong solution to Navier–Stokes equations converges to the Eulerian one in C([0, T]; H 1(Ω)) and ${L^\infty((0,T) \times \Omega)}$ , where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also. 相似文献
5.
We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields
are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This
toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided
the initial data belong to a “large” set in the Sobolev space H
1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the
class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions
of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show
how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear)
inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits
a short and simple proof of the existence of strong solutions for all time. 相似文献
6.
We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator. 相似文献
7.
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T
0 > 0, ν
0 > 0 and a unique continuous family of strong solutions u
ν
(0 ≤ ν < ν
0) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T
0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary
conditions imposed on curl
u
ν
and curl
2
u
ν
.
相似文献
8.
A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcys equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier–Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcys linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix. 相似文献
9.
In this paper we deal with the isentropic (compressible) Navier-Stokes equation in one space dimension and we adress the problem of the boundary controllability for this system. We prove that we can drive initial conditions which are sufficiently close to some constant states to those constant states. This is done under some natural hypotheses on the time of control and on the regularity on the initial conditions. 相似文献
10.
11.
Eduard Feireisl Antonín Novotný Yongzhong Sun 《Archive for Rational Mechanics and Analysis》2014,212(1):219-239
We show that any weak solution to the full Navier–Stokes–Fourier system emanating from the data belonging to the Sobolev space W 3,2 remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions. 相似文献
12.
Yong Wang 《Archive for Rational Mechanics and Analysis》2016,221(3):1345-1415
In the present paper, we study the uniform regularity and vanishing dissipation limit for the full compressible Navier–Stokes system whose viscosity and heat conductivity are allowed to vanish at different orders. The problem is studied in a three dimensional bounded domain with Navier-slip type boundary conditions. It is shown that there exists a unique strong solution to the full compressible Navier–Stokes system with the boundary conditions in a finite time interval which is independent of the viscosity and heat conductivity. The solution is uniformly bounded in \({W^{1,\infty}}\) and is a conormal Sobolev space. Based on such uniform estimates, we prove the convergence of the solutions of the full compressible Navier–Stokes to the corresponding solutions of the full compressible Euler system in \({L^\infty(0,T; L^2)}\), \({L^\infty(0,T; H^{1})}\) and \({L^\infty([0,T]\times\Omega)}\) with a rate of convergence. 相似文献
13.
14.
Reinhard Farwig Hermann Sohr Werner Varnhorn 《Journal of Mathematical Fluid Mechanics》2014,16(2):307-320
Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ . 相似文献
15.
We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction
conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary
layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains,
in both dimensions two and three. 相似文献
16.
Stefano Bosia Vittorino Pata James C. Robinson 《Journal of Mathematical Fluid Mechanics》2014,16(4):721-725
We give simple proofs that a weak solution u of the Navier–Stokes equations with H 1 initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L s (0, T;L r,∞(Ω)) or if its L s,∞(0, T;L r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1. 相似文献
17.
Jürgen Saal 《Journal of Mathematical Fluid Mechanics》2006,8(2):211-241
We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space
It is proved that the associated Stokes operator is sectorial and admits a bounded H∞-calculus on
As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations
with Robin boundary conditions. 相似文献
18.
New sufficient conditions of local regularity for suitable weak solutions to the non-stationary three-dimensional Navier–Stokes
equations are proved. They contain the celebrated Caffarelli–Kohn–Nirenberg theorem as a particular case.
相似文献
19.
20.
We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L 2 norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions. 相似文献