Regularity Criterion In Terms Research Articles

Regularity criterion in terms of the oscillation of pressure for the 3D Navier–Stokes equations

Regularity criterion in terms of the oscillation of pressure for the 3D Navier–Stokes equations

Regularity criterion in terms of BMO-type norm of the pressure gradient for the Navier-Stokes equations on unbounded domains

In this note we consider the initial-boundary value problem for the incompressible Navier-Stokes equations on unbounded domains in Rd by imposing the non-slip boundary condition. Then for locally uniform domains, introduced by Jones [20], we present a sufficient condition in terms of the BMO-type norm of the gradient of the pressure for the regularity of Leray-Hopf weak solutions.

Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier–Stokes equations in three dimensions. It is shown that the velocity is regular near a point z if its scaled $$L^p_tL^q_x$$ -norm of some quantities related to the velocity field is finite and the scaled $$L^p_tL^q_x$$ -norm of one velocity component is sufficiently small near z.

Navier–Stokes equations: regularity criteria in terms of the derivatives of several fundamental quantities along the streamlines—the case of a bounded domain

In this paper we deal with the conditional regularity of the weak solutions of the Navier–Stokes equations on a bounded domain endowed with Navier boundary conditions, Navier-type boundary conditions or Dirichlet boundary conditions. We prove the regularity criteria which are based on the directional derivatives of several fundamental quantities along the streamlines, namely the velocity magnitude, the kinetic energy, the pressure, the velocity field and the Bernoulli pressure. In striking contrast to the known criteria in which the mentioned quantities were differentiated along a fixed vector, our criteria are mostly optimal for the whole range of parameters and have a clear physical meaning.

On Regularity Criteria via Pressure for the 3D MHD Equations in a Half Space

In this paper, we give regularity criteria in terms of the magnetic pressure in Lorentz spaces.

Improved regularity criteria for the MHD equations in terms of pressure using an Orlicz norm

We present new regularity criteria in terms of the negative part of the pressure p or the positive part of the extended Bernoulli pressure B≔p+12|u|2+12|b|2, where u is the velocity, and b is the magnetic field. The criteria extend the previously known results, and the extension is enabled by the use of an appropriate Orlicz norm.

Mathematical reformulation of the Kolmogorov–Richardson energy cascade in terms of vortex stretching

In this paper, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov–Richardson energy cascade in terms of vortex stretching. By using the description, we prove that if the Navier–Stokes flow satisfies a new regularity criterion in terms of the enstrophy production rate, then the flow does not blow up. Our DNS results seem to support this regularity criterion. Next, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of the scale-locally of the vortex stretching/compressing (i.e. energy cascade) process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the −5/3 power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses.

Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

On the Serrin-Type Condition on One Velocity Component for the Navier–Stokes Equations

In this paper we consider the regularity problem of the Navier–Stokes equations in $$ {\mathbb {R}}^{3} $$ . We show that the Serrin-type condition imposed on one component of the velocity $$ u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))$$ with $$ \frac{2}{p}+ \frac{3}{q} <1$$ , $$ 3<q \le +\infty $$ implies the regularity of the weak Leray solution $$ u: {\mathbb {R}}^{3} \times (0,T) \rightarrow {\mathbb {R}}^{3} $$ , with the initial data belonging to $$ L^2({\mathbb {R}}^3) \cap L^3({\mathbb {R}}^{3})$$ . The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

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.

On the Deformation Tensor Regularity for the Navier\u2013Stokes Equations in Lorentz Spaces

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor mathcal {D}(u) to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm Vert lambda _{2}^{+}Vert _{L^{p,infty }(0,T; L ^{q,infty }(mathbb {R}^{3}))} with {2}/{p}+{3}/{q}=2( {3}/{2}<qle infty ) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

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 , ,

$$\varepsilon $$-Regularity criteria for the 3D Navier–Stokes equations in Lorentz spaces

In this paper, we are concerned with the regularity of suitable weak solutions to the 3D Navier–Stokes equations in Lorentz spaces. We obtain $$\varepsilon $$ -regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity or deformation tensor in Lorentz spaces, which generalizes the corresponding results derived by Gustafson et al. (Commun Math Phys 273:161–176, 2007) in Lebesgue spaces. As applications, this allows us to extend recent results involving Leray’s blow up rate in time, and to show that the number of singular points of weak solutions belonging to $$ L^{p,\infty }(-1,0;L^{q,l}({\mathbb {R}}^{3})) $$ is finite, where the pair (p, q) satisfies $$ {2}/{p}+{3}/{q}=1$$ with $$3<q<\infty $$ and $$q\le l <\infty $$ .

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).

A Regularity Criterion in Weak Spaces to Boussinesq Equations

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 0 , T ) . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.

Growth in the Muskat problem

We review some recent results on the Muskat problem modelling multiphase flow in porous media. Furthermore, we prove a new regularity criteria in terms of some norms of the initial data in critical spaces (Ẇ1,∞ and Ḣ3∕2).

Low modes regularity criterion for a chemotaxis-Navier-Stokes system

In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves certain existing criteria in the literature.

A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations

SummaryIn this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $t\in (0,T)$, the ‘large velocity’ region $\Omega:=\{(x,t)\mid |u(x,t)|&amp;gt;C(q)\left|\mkern-2mu\left|{u}\right|\mkern-2mu\right|_{L^{3q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.

A remark on the existence of a class of stretched [formula omitted] Magnetohydrodynamics flow

We consider the existence of a class of stretched solutions of 212D Magnetohydrodynamics equations in R3, which are sometimes called the columnar or two and half dimensional flows. The third components of the state variables have the form u3=x3γ1(x1,x2)+φ1(x1,x2) and B3=x3γ2(x1,x2)+φ2(x1,x2) and the first two components of the state variables depend on x1 and x2 only. We prove the local existence of such a flow in Sobolev spaces and give the regularity criteria in terms of 2D vorticity ω or 2D magnetic density j (but not both). Next, we identify the global existence of a class of axisymmetric flow without swirl for both velocity and magnetic field as a corollary of the main theorem. Finally, we present some exact global solutions as well as singular solutions with a special structure for viscous case and some exact global solutions for full inviscid case.