Stokes Equations In Terms Research Articles

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.

On the approximation of turbulent fluid flows by the Navier–Stokes-[formula omitted] equations on bounded domains

The Navier–Stokes-α equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant α is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when α=0, one recovers the classical Navier–Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier–Stokes-α equations can also be interpreted as a regularization of the Navier–Stokes equations, where α stands for the regularization parameter.In this paper we first present the Navier–Stokes-α equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier–Stokes-α equations to that of the solutions of the Navier–Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter α and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier–Stokes equations in terms of the L2 norm.

A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L2(0,T;Ḃ∞,∞−1). This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011).

Regularity Criteria for Weak Solutions to the Navier–Stokes Equations in Terms of Spectral Projections of Vorticity and Velocity

Regularity Criteria for Weak Solutions to the Navier–Stokes Equations in Terms of Spectral Projections of Vorticity and Velocity

Smoothed boundary method for simulating incompressible flow in complex geometries

Simulating flow through porous media with explicit considerations of complex microstructures is very challenging using conventional sharp-interface methods because of the difficulties in generating meshes conformal to complex geometries. In this work, a diffuse interface embedded boundary method known as the Smoothed Boundary Method (SBM) is utilized to facilitate simulations of fluid dynamics involving complex geometries. In diffuse-interface methods, the geometry is described by a domain parameter. The SBM allows the straightforward reformulation of the time-dependent Navier–Stokes equations in terms of this domain parameter, using only differentiation identities. Thus, enforcing the appropriate boundary conditions at the irregular embedded boundary is greatly simplified. Adaptive mesh refinement is used to increase the accuracy of the diffuse interface method by allowing a thinner interfacial thickness to be used in the domain parameter. The SBM-formulated Navier–Stokes equations are solved with the Finite Difference Method on refined mesh systems. Sharp-interface Finite Element Method simulations using the commercial software COMSOL on body-conforming meshes are also provided for comparison. Favorable agreement between the two methods is observed. Since it is no longer necessary for the mesh to conform to the complex geometry, the grid system for the SBM simulations can be generated rapidly and without additional manual interventions, making the entire simulation process more expedient.

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

We deal with the conditional regularity of the weak solutions to the Navier–Stokes equations. We discuss a famous criterion by Vasseur in terms of and extend this criterion to bounded domains with Navier and Navier‐type boundary conditions. Inspired by the equality , we further prove an optimal regularity criterion in terms of both for the whole three‐dimensional space and bounded domains with Navier's, Navier‐type, and Dirichlet boundary conditions. It specially means for that the control of the energy flow in the critical norms provides the regularity of solutions. This criterion is proved by two different techniques.

Local regularity conditions on initial data for local energy solutions of the Navier–Stokes equations

We study the regular sets of local energy solutions to the Navier–Stokes equations in terms of conditions on the initial data. It is shown that if a weighted \(L^2\) norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli et al. (Comm. Pure Appl. Math. 35(6):771–831, 1982) and our recent paper (Kang et al., Pure Appl. Anal. arXiv:2006.13145) as well.

Regularity criteria for 3D Navier–Stokes equations in terms of a mid frequency part of velocity in B˙-1 ∞,∞

Abstract We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular in ( 0 , T ] {(0,T]} if the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , the mid frequency part of Fourier modes k 2 ≤ | ξ | < k {\frac{k}{2}\leq|\xi|<k} , is small depending on the kinematic viscosity ν, initial value u 0 {u_{0}} and the maximum of an averaged energy dissipation rate A ≡ sup t ∈ ( 0 , T ) ⁡ ( ν ⁢ t - 1 ⁢ ∫ 0 t ∥ ∇ ⁡ u ∥ 2 ⁢ 𝑑 τ ) A\equiv\sup_{t\in(0,T)}\biggl{(}\nu t^{-1}\int_{0}^{t}\lVert\nabla u\rVert^{2}% \,d\tau\biggr{)} for some k ≥ k 0 ⁢ ( ν , u 0 , A ) {k\geq k_{0}(\nu,u_{0},A)} . In particular, when a sufficiently high frequency part of u 0 {u_{0}} decays fast at an exponential rate, then we obtain regularity conditions in terms of smallness of the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , which involve only the known data ν and u 0 {u_{0}} .

Fractional Navier–Stokes regularity criterion involving the positive part of the intermediate eigenvalue of the strain matrix

This paper proves the regularity criterion for the solutions of the 3D fractional Navier–Stokes equations in terms of the positive part of the intermediate eigenvalue of the strain matrix, which can be regarded as an improvement and extension of the results obtained by Miller [Arch. Ration. Mech. Anal. 235, 99–1139 (2020)] and by Wu [Evol. Equations Control Theory 10, 511–518 (2021)].

Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field

We establish some regularity criteria for the solutions to the Navier–Stokes equations in the full three-dimensional space in terms of one directional derivative of the velocity field. Revising the method used by Zujin Zhang (2018), we show that a weak solution u is regular on (0, T] provided that ∂u∂x3∈Lp(0,T;Lq(R3)) with s=2 for 3≤q≤6, 116<s≤2 for 6≤q≤66s−11 where s=2p+3q. They improve the known results 2p+3q=32 for 2≤q≤∞, 2p+3q≤85+911q for 52≤q<∞ and 2p+3q≤1411+35q for 4≤q<∞ .

Blow up criterion for the 2D full compressible Navier–Stokes equations involving temperature in critical spaces

In this paper, we derive some new blow up criterion for the 2D full compressible Navier–Stokes equations in terms of the density and the temperature in critical spaces. This is an improvement of corresponding results obtained by Wang and Zhang [Appl. Math. Comput. 232, 719–726 (2014)].

Locally Space-Time Anisotropic Regularity Criteria for the Navier–Stokes Equations in Terms of Two Vorticity Components

In this paper we prove the regularity of Leray weak solutions of the Navier–Stokes equations as long as the vorticity projection to a plane is bounded in the scale critical space $$L^p(0,T;L^q)$$ , $$2/p+3/q=2$$ , $$q \in (3/2,\infty )$$ . The plane may vary in space and time while the unit vector $$v=v(x,t)$$ orthogonal to the plane is locally a Holder function in space with the coefficient 1/2. This extends previous works by Chae and Choe and by Miller. We further show that a generalized form of this criterion improves several other regularity criteria in terms of the vorticity direction known from the literature.

A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity

In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space L t 4 L x 2 L^4_t L^2_x , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.

Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in [formula omitted

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L2(0,T;BMO−1). We also prove that u can be smoothly extended beyond T, if the horizontal gradient of horizontal components of velocity u belong to L2(0,T;BMO−1). The proof relies on the technique of the Bony decomposition and some Fourier multiplier theorems.

Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity

In this paper, we consider the 3D Navier–Stokes equations in the whole space. We investigate some new inequalities and a priori estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely, , , . Moreover, we extend the range of q while the solution is axisymmetric, that is, the axisymmetric solution u is regular in (0, T], if , ,

A regularity criterion for the 3D Boussinesq equations

ABSTRACT In this paper, we consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative of the velocity. Acta Appl Math. 2015;140(1):1–9; Gala S, Ragusa M A. On the regularity criterion for the Navier–Stokes equations in terms of one-directional derivative. Asian-Eur J Math. 2017;10(1):1750012).

An optimal regularity criterion for the Navier–Stokes equations proved by a blow-up argument

We study the conditional regularity for the incompressible Navier–Stokes equations in terms of one directional derivative of the velocity field. We show that if u is a weak solution in the whole three dimensional space and ∂3u∈Lp(0,T;Lq(R3)), T>0, where 2∕p+3∕q=2 and q∈(3,19∕6], then u is regular on (0,T], crossing so for the first time the barrier q=3. The proof is based on a suitable combination of a blow-up argument and mutual estimates of certain integral quantities pertained to the rescaled solution.

The end-point regularity criterion for the Navier–Stokes equations in terms of [formula omitted

We show the following extension of the recent optimal regularity results by Kukavica & Ziane, Cao and Zhang: If u is a solution of the Navier–Stokes equations in the whole space R3 and ∂3u∈Lp(0,T;Lq(R3)), T>0, where 2∕p+3∕q=2 and q∈(3∕2,3], then u is regular on (0,T].

A relaxed upper and lower triangular splitting preconditioner for the linearized Navier–Stokes equation

Based on the block triangular splitting of the original coefficient matrix, a relaxed upper and lower triangular splitting (RULT) preconditioner for the linearized incompressible Navier–Stokes equations is considered. The convergence analysis of the RULT iteration method is presented. The spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed, respectively. Then the quasi-optimal parameters of the RULT preconditioner are derived. Finally, some numerical results are carried out to show the effectiveness of the RULT preconditioner, and verify that the RULT preconditioner outperforms the MDS, GRS, MRS and RBTS preconditioners for the GMRES method for solving the linearized incompressible Navier–Stokes equation in terms of the number of iterations and computation times.

The anisotropic regularity criteria for 3D Navier–Stokes equations involving one velocity component

The anisotropic regularity criterion for the Navier–Stokes equations in terms of the one component of the velocity gradient (∂iuj,1≤i,j≤3) is investigated. It is an improvement of results of Qian (2016) and is also a complement of the results of Guo et al. (2018) for i=j. Besides, the consistent regularity criterion without distinguishing i=j and i≠j is also obtained. Finally, the regularity result based on the fractional derivatives for one component of the velocity field Dhαu3 with α∈[0,1] is also established, which can be reduced to some previous results with α=0 and α=1.