Suitable Weak Solutions Research Articles

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

On the interior regularity conditions of a suitable weak solution to 3D MHD equations

The Minkowski dimension of suitable weak solutions of the 3D co-rotational Beris-Edwards system

The Minkowski dimension of suitable weak solutions of the 3D co-rotational Beris-Edwards system

On the existence of solutions to generalized Navier–Stokes–Fourier system with dissipative heating

AbstractWe consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier–Stokes–Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck–Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck–Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

Remarks on interior regularity criteria without pressure for the Navier-Stokes equations

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the LtpLxq(Q1) norm of the velocity is sufficiently small, where 1≤2p+3q<2 and 2≤p≤∞. It improves the recent result of p,q>2 by Kwon [15], and also generalizes Chae-Wolf's Lt∞Lx32+ criterion [3].

Parabolic turbulence k-epsilon model with applications in fluid flows through permeable media

In this work, we study a one-equation turbulence \(k\)-epsilon model that governs fluid flows through permeable media. The model problem under consideration here is derived from the incompressible Navier-Stokes equations by the application of a time-averaging operator used in the \(k\)-epsilon modeling and a volume-averaging operator that is characteristic of modeling unsteady porous media flows. For the associated initial- and boundary-value problem, we prove the existence of suitable weak solutions (average velocity field and turbulent kinetic energy) in the space dimensions of physics interest.

Global existence of suitable weak solutions to the 3D chemotaxis-Navier-Stokes equations

Global existence of suitable weak solutions to the 3D chemotaxis-Navier-Stokes equations

Partial regularity of suitable weak solution to a three-dimensional fractional parabolic-elliptic chemotaxis-Navier–Stokes system

In this paper, we investigate a fractional parabolic-elliptic chemotaxis-Navier–Stokes system in spatial dimensions three and obtain the global existence of the suitable weak solution by a contraction mapping theorem. Furthermore, we improve the regularity of the solution through a local maximal L p regularity estimate for the fractional heat equation such that the suitable weak solution is smooth away from a closed set whose one-dimensional parabolic Hausdorff measure is zero, which extends the partial regularity theory of Caffarelli, Kohn and Nirenberg [] on the Navier–Stokes equation to the fractional parabolic-elliptic chemotaxis-Navier–Stokes system.

Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

The role of the pressure in the regularity theory for the Navier-Stokes equations

We first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak solutions. Then, an $\varepsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\varepsilon$-regularity criterion with a new approach using a local version of Leray projection operator. As an application of the approach, we obtain the short-time regularity result on a bounded domain for dissipative solutions.

Fractal dimension of potential singular points set in the Navier–Stokes equations under supercritical regularity

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q&lt;\infty$ and $2&lt; p&lt;\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

Navier–Stokes equations: a new estimate of a possible gap related to the energy equality of a suitable weak solution

The paper is concerned with the IBVP of the Navier-Stokes equations. The result of the paper is in the wake of analogous results obtained by the authors in previous articles Crispo et al. (Ricerche Mat 70:235–249, 2021). The goal is to estimate the possible gap between the energy equality and the energy inequality deduced for a weak solution.

A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions

Abstract The Cahn–Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for this equation have been proposed. Recently, a family of models using Cahn–Hilliard-type equations on the boundary of the domain to describe adsorption processes was analysed (cf. Knopf, P., Lam, K. F., Liu, C. &amp; Metzger, S. (2021) Phase-field dynamics with transfer of materials: the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. ESAIM: Math. Model. Numer. Anal., 55, 229–282). This family of models includes the case of instantaneous adsorption processes studied by Goldstein, Miranville and Schimperna (2011, A Cahn–Hilliard model in a domain with non-permeable walls. Phys. D, 240, 754–766) as well as the case of vanishing adsorption rates, which was investigated by Liu and Wu (2019, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal., 233, 167–247). In this paper, we are interested in the numerical treatment of these models and propose an unconditionally stable, linear, fully discrete finite element scheme based on the scalar auxiliary variable approach. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. Thereby, when passing to the limit, we are able to remove the auxiliary variables introduced in the discrete setting completely. Finally, we present simulations based on the proposed linear scheme and compare them to results obtained using a stable, nonlinear scheme to underline the practicality of our scheme.

Local $ \varepsilon $-regularity criteria for the five dimensional stationary Navier-Stokes equations

We establish interior and boundary $ \varepsilon $-regularity criteria at one scale for suitable weak solutions to the five dimensional stationary incompressible Navier-Stokes equations which improve previous results in [19] and [32]. Our proof is based on an iteration argument, Campanato's method, and interpolation techniques.

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

On the interior regularity criterion for suitable weak solutions to the 3d magneto-hydrodynamics equations in terms of the velocity gradient

A remark on weak-strong uniqueness for suitable weak solutions of the Navier–Stokes equations

We extend Barker’s weak-strong uniqueness results for the Navier–Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces.

The Hele–Shaw flow as the sharp interface limit of the Cahn–Hilliard equation with disparate mobilities

In this paper, we study the sharp interface limit for solutions of the Cahn–Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn–Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele–Shaw flow with optimal energy-dissipation rate.

Infinitesimal invariance of completely Random Measures for 2D Euler Equations

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.

Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation

We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation −Δs in the critical case of s=32, which satisfy certain local energy inequalities and whose singular sets have a locally finite two-dimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for s=32. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when s=32, the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentration-compactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.

Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫B1|u(x)|3dx+∫B1|f(x)|qdx or ∫B1|∇u(x)|2dx+∫B1|∇u(x)|2dx(∫B1|u(x)|dx)2+∫B1|f(x)|qdx with q>3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫B1+|u(x)|3dx+∫B1+|f(x)|3dx or ∫B1+|∇u(x)|2dx+∫B1+|f(x)|3dx is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8], Indiana Univ. Math. J., 2012), Dong-Gu ([7], J. Funct. Anal., 2014), and Liu-Wang ([27], J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary.

Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities

We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier–Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. In the present work, we devise for the resulting evolution problem a suitable weak solution concept based on the framework of varifolds and establish as the main result a weak-strong uniqueness principle in 2D. The proof is based on a relative entropy argument and requires a non-trivial further development of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech. Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects of the necessarily singular geometry of the evolving fluid domains, we work for simplicity in the regime of same viscosities for the two fluids.