Suitable Weak Solutions Research Articles

Coupling Rate-Independent and Rate-Dependent Processes: Existence Results

We address the analysis of an abstract system coupling a rate-independent process with a rate-dependent nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in viscoelastic solids with inertia, and to a recently proposed model for damage with plasticity.

Suitable weak solutions to the 3D Navier–Stokes equations are constructed with the Voigt approximation

In this paper we consider the Navier–Stokes equations supplemented with either the Dirichlet or vorticity-based Navier slip boundary conditions. We prove that weak solutions obtained as limits of solutions of the Navier–Stokes–Voigt model satisfy the local energy inequality, and we also prove certain regularity results for the pressure. Moreover, in the periodic setting we prove that if the parameters are chosen in an appropriate way, then we can construct suitable weak solutions through a Fourier–Galerkin finite-dimensional approximation in the space variables.

A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [ 1 ], as a corollary, under suitable assumptions of local character on the initial data, we investigate the behavior in time of the \begin{document}$L_{loc}^\infty$\end{document} -norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator.

On the geometric regularity conditions for the 3D Navier–Stokes equations

We prove geometrically improved version of Prodi–Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier–Stokes equations and denote {f}+=max{f,0}, QT=R3×(0,T). If {(v×ω∣ω∣)⋅Λβv∣Λβv∣}+∈Lx,tγ,α(QT) with 3/γ+2/α≤1 for some γ>3 and 1≤β≤2, then the local smooth solution v of the Navier–Stokes equations on (0,T) can be continued to (0,T+δ) for some δ>0. We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier–Stokes equations in a space–time domain containing z0=(x0,t0), let Qz0,r=Bx0,r×(t0−r2,t0) be a parabolic cylinder in the domain. We show that if either {(v×ω∣ω∣)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤1, or {(v∣v∣×ω)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤2, (γ≥2, α≥2), then z0 is a regular point for v. This improves previous local regularity criteria for the suitable weak solutions.

Dimension Reduction for the Full Navier–Stokes–Fourier system

It is well known that the full Navier-Stokes-Fourier system does not possess a strong solution in three dimensions which causes problems in applications. However, when modeling the flow of a fluid in a thin long pipe, the influence of the cross section can be neglected and the flow is basically one-dimensional. This allows us to deal with strong solutions which are more convenient for numerical computations. The goal of this paper is to provide a rigorous justification of this approach. Namely, we prove that any suitable weak solution to the three-dimensional NSF system tends to a strong solution to the one-dimensional system as the thickness of the pipe tends to zero.

Refined regularity class of suitable weak solutions to the 3D magnetohydrodynamics equations with an application

By means of blow-up method and the special structure of the 3D viscous magnetohydrodynamics equations, we derive some interior regularity criteria in terms of horizontal part of the velocity with sufficiently small local scaled norm and both the vertical part of the velocity and the magnetic field with bounded local scaled norm for the suitable weak solutions to this system. As an application, this allows us to improve the previous limiting case for the regularity criterion about the MHD equations.

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.

Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations

In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in [Formula: see text] for [Formula: see text]. In comparison with the work of the 3D fractional Navier–Stokes equations obtained by Tang and Yu in [Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations, Comm. Math. Phys. 334 (2015) 1455–1482], our results include their endpoint case [Formula: see text] and the external force belongs to a more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.

An Anisotropic Partial Regularity Criterion for the Navier–Stokes Equations

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

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.

Partial regularity of solutions to the fractional Navier-Stokes equations

We study the partial regularity of suitable weak solutions to the Navier-Stokes equations with fractional dissipation $\sqrt{-\Delta}^s$ in the critical case of $s=\frac{3}{2}$. We show that the two dimensional Hausdorff measure of space-time singular set of these solutions is zero.

The Minkowski dimension of interior singular points in the incompressible Navier–Stokes equations

We study the possible interior singular points of suitable weak solutions to the three dimensional incompressible Navier–Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set, which is bounded by 95/63. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty.

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.

Global existence and uniqueness of solutions to the three-dimensional Boussinesq equations

In this paper, we study the three-dimensional Boussinesq equations and obtain the global existence and uniqueness of a suitable weak solution in unbounded exterior domain.

Local regularity criteria of the 3D Navier–Stokes and related equations

In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian (−Δ)α (3/4≤α<1). In the first part, we derive some ϵ-regularity criteria in terms of the deformation tensor D(u) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u, the gradient of the velocity ∇u, the rotation tensor Ω(u) and the deformation tensor D(u). This generalizes the results obtained by Gustafson et al. (2007).

On the spatial asymptotic decay of a suitable weak solution to the Navier–Stokes Cauchy problem

We prove space–time decay estimates of suitable weak solutions to the Navier–Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result obtained in [] for the behavior of the solution in a neighborhood of t = 0 in the -norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of for . Following Leray’s point of view, roughly speaking our result proves that a possible space–time turbulence does not perturb the asymptotic spatial behavior of the initial data of a suitable weak solution.

Blow-up of Critical Besov Norms at a Potential Navier–Stokes Singularity

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 <p,q< \infty$, gives rise to a strong solution with a singularity at a finite time $T>0$, then the norm of the solution in that Besov space becomes unbounded at time $T$. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in $L^3(\mathbb{R}^3)$. Our proof uses profile decompositions and is based on our previous work (Math. Ann. 355(4):1527--1559, 2013) which provided an alternative proof of the $L^3(\mathbb{R}^3)$ result. For very large values of $p$, an iterative method, which may be of independent interest, enables us to use some techniques from the $L^3(\mathbb{R}^3)$ setting. (To appear in Communications in Mathematical Physics.)

Partial Hölder regularity of the steady fractional Navier–Stokes equations

In this paper, we study steady suitable weak solutions to the fractional Navier–Stokes system in \(\mathbb {R}^3\) with \(1/2 < s < 1\), where s denotes the power of the negative Laplacian. We show that if \(1/2<s<5/6\), any steady suitable weak solution is regular away from a relatively closed set with zero \((5-6s)\)-Hausdorff measure and it is regular when \(5/6\le s<1\).

Regularity of solutions to axisymmetric Navier–Stokes equations with a slightly supercritical condition

Consider an axisymmetric suitable weak solution of 3D incompressible Navier–Stokes equations with nontrivial swirl, v=vrer+vθeθ+vzez. Let z denote the axis of symmetry and r be the distance to the z-axis. If the solution satisfies a slightly supercritical assumption (that is, |v|≤C(ln⁡|ln⁡r|)αr for α∈[0,0.028] when r is small), then we prove that v is regular. This extends the results in [6,16,18] where regularities under critical assumptions, such as |v|≤Cr, were proven.As a useful tool in the proof of our main result, an upper-bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper.

On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

We investigate the size of the regular set for suitable weak solutions of the Navier--Stokes equation, in the sense of Caffarelli--Kohn--Nirenberg. 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 $\{t>0\}$.