A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces

For the incompressible Navier–Stokes equations in R³${\mathbb{R}}^3$, a regularity criterion for weak solutions is proved under the assumption that the pressure belongs to the scaling invariant Lorentz space with small norm, while corresponding results for the velocity field were proved by Sohr. The main theorem continues and extends a previous result given by the author.     

On the uniqueness of weak solutions for the 3D Navier–Stokes equations

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.     

On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains

Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R ³. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg in R ³ and Sohr-von Wahl in exterior domains to general domains.     

On regularity of weak solutions to the instationary Navier–Stokes system: a review on recent results

Consider a weak instationary solution \(u\) of the Navier–Stokes equations in a domain \(\varOmega \subsetneq \mathbb {R}^3\) with Dirichlet boundary data \(u=0\) on \(\partial \Omega \) , i.e., \(u\) solves the Navier–Stokes system in the sense of distributions and $$\begin{aligned} u \in L^\infty \left( 0,T;L^2(\varOmega )\right) \cap L^2 \left( 0,T;W^{1,2}_0(\varOmega )\right) . \end{aligned}$$ Since the pioneering work of J. Leray 1933/34 it is an open problem whether weak solutions are unique and smooth. The main step—to nowadays knowledge—is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^s \left( 0,T;L^q(\varOmega )\right) \) where \(s>2, q>3\) and \(\frac{2}{s}+ \frac{3}{q}=1\) . This review reports on recent progress in this important problem, considering this issue locally on an initial interval \([0,T')\) , \(T'<T\) , i.e., the problem of optimal initial values \(u(0)\) , globally on \([0,T)\) , and from a one-sided point of view \(u \in L^s \left( T'-\varepsilon ,T';L^q(\varOmega )\right) \) or \(u \in L^s\left( T',T'+\varepsilon ;L^q(\varOmega )\right) \) . Further topics deal with the energy (in-)equality, uniqueness of weak solutions, blow-up phenomena and the analysis in critical spaces for the whole space case.     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.     

A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

In this paper, we obtain the global existence of weak solutions to the compressible quantum Navier–Stokes equations. By virtue of a useful identity and an interesting estimate, we solve the critical case that the viscosity equals the dispersive coefficient. This result removes the restrictions on the coefficients and improves the recent work of Antonell and Spirito (2017) in some senses.     

On local regularity for suitable weak solutions of the Navier–Stokes equations near the boundary

A class of sufficient conditions for local boundary regularity of suitable weak solutions of nonstationary three-dimensional Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals, which are invariant with respect to the scaling of the Navier–Stokes equations. Bibliography: 27 titles.     

Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain

Archiv der Mathematik - We give some regularity criterion (of a weak $$-L^{p}$$ Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain $$\Omega \subset \mathbb...     

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with ${r \in [1, \infty)}In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with r ? [1, ￥){r \in [1, \infty)} , which are beyond Serrin’s condition.     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {Omega_0} subset bigcuplimits_{k = 1}^N {{Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {left| {{v_r}} right|_{{L_infty }left( {0,T;{L_3}left( {{Omega_k}} right)} right)}} leq {alpha_1},or;{left| {frac{{{v_r}}}{r}} right|_{{L_infty }left( {0,T;{L_{3/2}}left( {{Omega_k}} right)} right)}} leq {alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) left( {v in W_2^{2,1}left( {Omega times left( {0,T} right)} right),nabla p in {L_2}left( {Omega times left( {0,T} right)} right)} right) . Bibliography: 28 titles.     

A new regularity criterion for weak solutions to the Navier–Stokes equations

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier–Stokes equations. We show that if any one component of the velocity field belongs to $L^{\alpha }([0,T);L^{\gamma }({\mathbb{R}}^3))$ with $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.⩽\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $6<\gamma ⩽\infty$, then the weak solution actually is regular and unique.     

On very weak solutions of the stationary Navier–Stokes equations

We study the stationary Navier–Stokes equations in a bounded domain Ω of R ³ with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) to obtain solvability results for the Navier–Stokes equations with very irregular data. In this article, we prove a complete solvability result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono (Preprint).     

On Landau’s solutions of the Navier–Stokes equations

A special class of solutions of the n-dimensional steady-state Navier–Stokes equations is considered. Bibliography: 23 titles.     

On symmetric Leray solutions of the stationary Navier–Stokes equations

A classical result of Amick (Acta Math 161:71–130, 1988) on the nontriviality of the symmetric Leray solutions of the steady-state Navier–Stokes equations in the plane is extended to Lipschitz domains. This results is compared with the famous Stokes paradox of linearized hydrodynamics and applied to a mixed problem of some interest in the applications.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.