No Arabic abstract
We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure $p$ appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint $mathrm{div}, u =0$.
In this paper we address the large-scale regularity theory for the stationary Navier-Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier-Stokes equations. We prove: (i) a large-scale Calderon-Zygmund estimate, (ii) a large-scale Lipschitz estimate, (iii) large-scale higher-order regularity estimates, namely, $C^{1,gamma}$ and $C^{2,gamma}$ estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale $C^{1,gamma}$ regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale $C^{2,gamma}$ regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale.
We study the stationary nonhomogeneous Navier--Stokes problem in a two dimensional symmetric domain with a semi-infinite outlet (for instance, either parabo-loidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value ${bf a}$ over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<infty$ and $lim_{tto T}sqrt{T-t}||u(t)||_{L^infty}=infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $uin L^{2,infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,infty}([0,T];BMO(Omega))}<infty$ for arbitrary $Omegasubseteqmathbb{R}^3$ then the local energy equality is valid on $[0,T]timesOmega$. As a corollary, we also prove $||u||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}<infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)||_{BMO}$ must blow up at a rate faster than $frac{c}{sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}=M<infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h||_{L^{2,infty}([0,T];BMO(mathbb{R}^3))}leq c_M$ then $u$ is regular on $(0,T]timesmathbb{R}^3$.
By using a set of scaling limits, the authors in cite{ADFL,SS} proposed a framework of deriving the Hall-MHD equations from the two-fluids Euler-Maxwell equations for electrons and ions. In this paper, we derive the Hall-MHD equations from the Navier-Stokes-Maxwell equations with generalized Ohms law in a mathematically rigorous way via the spectral analysis and energy methods.