Leray Hopf Weak Solutions Research Articles

The Navier–Stokes equations with body forces decaying coherently in time

The long-time behavior of solutions of the three-dimensional Navier–Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time t tends to infinity, in a coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of et, t, ln⁡t, or other iterated logarithmic functions of t. We prove that all Leray–Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey–Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier–Stokes equations.

Energy equality for the 3D inhomogeneous Navier–Stokes equations in Lorentz–Besov spaces

We establish the energy equality for Leray–Hopf weak solutions of the 3D incompressible inhomogeneous Navier–Stokes equations in Lorentz–Besov spaces. This result may be regarded as an extension of that by Cheskidov and Luo (2020).

Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to Lloc1([0,+∞);Lexp(Rd;Rd×d)), where Lexp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray–Hopf weak solutions of the Navier–Stokes equations.

Partial regularity of Leray–Hopf weak solutions to the incompressible Navier–Stokes equations with hyperdissipation

Partial regularity of Leray–Hopf weak solutions to the incompressible Navier–Stokes equations with hyperdissipation

Remarks on the “Onsager Singularity Theorem” for Leray–Hopf Weak Solutions: The Hölder Continuous Case

In this paper, we first present an overview of the results related to energy conservation in spaces of Hölder-continuous functions for weak solutions to the Euler and Navier–Stokes equations. We then consider families of weak solutions to the Navier–Stokes equations with Hölder-continuous velocities with norms uniformly bound in terms of viscosity. We finally provide the proofs of our original results that extend the range of allowed exponents for inviscid limits producing solutions to the Euler equations satisfying the energy equality, and improve the so-called “Onsager singularity” theorem.

Global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity

This paper concerns the global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity. Motivated by the works of Tan and Wu in [arXiv:2111.13547v2] and Wu in [J. Math. Fluid Mech. 24, 111 (2022)], we establish the energy balance for a distributional solution in whole spaces Rd(d ? 2) provided that u, b ? L4L4 and ?b ? L8/3 L8/3. Moreover, as a corollary, we also obtain the energy conservation criterion for a Leray-Hopf weak solution.

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.

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

In this paper, we consider the analysis of a nudging based algorithm for data assimilation for the three-dimensional Boussinesq system, which we call the AOT system. A rigorous analysis of this algorithm for well-posed dissipative partial differential equations was first provided by Azouani, Olson and Titi (J Nonlinear Sci 24:277–304, 2014); thus justifying our terminology for the associated nudging system. We provide a sufficient condition, based solely on the observed velocity data obtained from a Leray-Hopf weak solution, for the global well-posedness, regularity and most crucially, the asymptotic tracking property of solutions of the associated (three-dimensional) nudging system. It is to be noted that neither regularity nor any knowledge of a uniform \({\mathbb {H}}^1\)-norm bound is a priori assumed on the solution of the original three-dimensional Boussinesq system from which the observations are obtained. As a corollary of our result, we obtain a novel observable regularity criterion based on finitely many observational data. Our condition also guarantees the construction of the Lipschitz continuous determining map, which is known to play a crucial role in the construction of the so-called determining form and in statistical data assimilation.

A boundary layer problem in nonflat domains with measurable viscous coefficients

AbstractWe study the convergence of weak solutions of the Navier–Stokes equations with vanishing measurable viscous coefficients in domains with nonflat boundaries. Sufficient anisotropic conditions on the vanishing rates of the viscous coefficients are found to prove the convergence of Leray–Hopf weak solutions of the Navier–Stokes equations to solutions of the corresponding Euler equations. As the domains are not flat, we apply a change of variables to flatten the domains. We then construct explicit boundary layers for the system of Navier–Stokes equations in the upper‐half space with measurable viscous coefficients. The result is new even when the viscous coefficients are constant, and it recovers the classical results when domains are flat and with constant viscous coefficients.

Energy equality in the isentropic compressible Navier-Stokes equations allowing vacuum

It is well-known that a Leray-Hopf weak solution in L4(0,T;L4(T3)) for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This further allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations, which is an improvement of corresponding results obtained by Yu in [31, Arch. Ration. Mech. Anal., 225 (2017)] and answers a question posed by Liang in [18, Proc. R. Soc. Edinb., Sect. A, Math. (2020)].

More on Convergence of Chorin’s Projection Method for Incompressible Navier–Stokes Equations

Kuroki–Soga (Numer. Math. 146:401–433, 2020) proved that Chorin’s fully discrete finite difference projection method, originally introduced by Chorin (Math. Comput. 23:341–353, 1969), is unconditionally solvable and convergent within an arbitrary fixed time interval to a Leray–Hopf weak solution of the incompressible Navier–Stokes equations on a bounded domain with an arbitrary external force. This paper is a continuation of Kuroki–Soga’s work to further exhibit mathematical aspects of the method. We show time-global solvability and convergence of the scheme; \(L^2\)-error estimates for the scheme in the class of smooth exact solutions; application of the method to the problem with a time-periodic external force to investigate time-periodic (Leray–Hopf weak) solutions, long-time behaviors, error estimates, etc.

On the low Mach number limit for 2D Navier–Stokes–Korteweg systems

<abstract><p>This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.</p></abstract>

On Masuda uniqueness theorem for Leray–Hopf weak solutions in mixed-norm spaces

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L∞L3 Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest.

Uniform stability for the 3D magneto-hydrodynamics equations

In this note, we are concerned with uniform stability for a Leray–Hopf weak solution of the MHD equations and obtain that the Leray–Hopf weak solution is uniformly stable with respect to a small perturbation of initial velocity, magnetic and external forcing. This extends and improves the results given by Gallagher–Planchon and Dong–Jia for Navier–Stokes equations.

On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations

We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.

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.

On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

In this paper we derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions $v$ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation $\pi\cong|v|^2$, where $\pi$ denotes the fluid pressure and $v$ the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem). It is shown that if $\f{\pi}{(e^{-|x|^2}+|v|)^{\theta}}\in L^p(0,T;L^{q,\infty})\,,$ where $0\leq\theta\leq1$ and $\f2p+\f3q=2-\theta$, then $v$ is regular on $(0,T]$. Note that, if $\Om$ is periodic, we may replace $\,e^{-|x|^2} \,$ by a positive constant. This result improves a 2018 statement obtained by one of the authors. Furthermore, as an integral part of our contribution, we give an overview on the known results on the P-V problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin (L-P-S) type.

Mixed-norm Lp-estimates for non-stationary Stokes systems with singular VMO coefficients and applications

We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli type inequalities for the Stokes systems with variable coefficients. A new ϵ-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regularity criterion.

A Uniqueness Result for 3D Incompressible Fluid-Rigid Body Interaction Problem

We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy the Prodi-Serrin $$\text {L}^{r}-\text {L}^{s}$$ condition are unique in the class of Leray-Hopf weak solutions.

Asymptotic Expansions for the Lagrangian Trajectories from Solutions of the Navier–Stokes Equations

Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.