Journal of Mathematical Fluid Mechanics最新文献

In this paper, we study weak solutions to the steady (time-independent) fractional Navier-Stokes system in (mathbb {R}^n). We offer a novel perspective to study the partial regularity of steady problem, and show that if (alpha in (frac{n+1}{6},frac{n+2}{6})), the Hausdorff dimension of singular set for the steady weak solution is at most (n+2-6alpha ). Our approach is inspired by the ideas of Katz and Pavlović (Geom. Funct. Anal. 12:2 (2002), 355-379) and Ożański (Anal. PDE 16:3 (2023)). This is the first attempt to apply the method of Katz and Pavlović to a steady setting.

The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power r in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power (r ge frac{1}{2}) and for fEV when (r>frac{5}{6}). Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter (alpha rightarrow 0). Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity (nu rightarrow 0) as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both (alpha , nu rightarrow 0). Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by m, is large compared to the perturbations. It was proved by Jiang–Jiang that the highest-order derivatives of the velocity increase with m and the convergence rate of the nonlinear system towards a linearized problem is of (m^{-1/2}) in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with m, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to m does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or m approaches infinity. Notably, our analysis reveals that the convergence rate in m is faster compared to the finding of Jiang–Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.

This paper deals with the uniqueness of mild solutions to the forced or unforced Navier-Stokes equations in the whole space. It is known that the uniqueness of mild solutions to the unforced Navier-Stokes equations holds in (L^{infty }(0,T;L^d({mathbb {R}}^d))) when (dge 4), and in (C([0,T];L^d({mathbb {R}}^d))) when (dge 3). As for the forced Navier-Stokes equations, when (dge 3) the uniqueness of mild solutions in (C([0,T];L^{d,infty }({mathbb {R}}^d))) with force f and initial data (u_{0}) in appropriate Lorentz spaces is known. In this paper we show that for (dge 3), the uniqueness of mild solutions to the forced Navier-Stokes equations in ( C((0,T];{widetilde{L}}^{d,infty }({mathbb {R}}^d))cap L^beta (0,T;{widetilde{L}}^{d,infty }({mathbb {R}}^d))) for (beta >2d/(d-2)) holds when there is a mild solution in (C([0,T];{widetilde{L}}^{d,infty }({mathbb {R}}^d))) with the same initial data and force. Here ({widetilde{L}}^{d,infty }) is the closure of ({L^{infty }cap L^{d,infty }}) with respect to (L^{d,infty }) norm.

We show that the collective effect of N rigid bodies ((mathcal {S}_{n,N})_{n=1}^N) of diameters ((r_{n,N})_{n=1}^N) immersed in an incompressible non–Newtonian fluid is negligible in the asymptotic limit (N rightarrow infty ) as long as their total packing volume (sum _{n=1}^N r_{n,N}^d), (d=2,3) tends to zero exponentially – ({sum _{n=1}^N r_{n,N}^d approx A^{-N}}) – for a certain constant (A > 1). The result is rather surprising and in a sharp contrast with the associated homogenization problem, where the same number of obstacles can completely stop the fluid motion in the case of shear thickening viscosity. A large class of non–Newtonian fluids is included, for which the viscous stress is a subdifferential of a convex potential.

We consider the Cauchy problem for the compressible Navier–Stokes equations in whole space and low Mach number limit problem. In this paper, we show that the incompressible part of the velocity strongly converges to the solution of the incompressible Navier–Stokes equations as the Mach number goes to 0 in the scaling critical space. We also show that the density and the compressible part of the velocity vanish. Moreover, we derive the diverging of the time derivative of the compressible part of the velocity as Mach number goes to 0. The proofs are based on the (L^1)-Maximal regularity for the heat equations and the Strichartz estimates for the wave equations.

Lagrangian variables are used to develop an explicit description of nonlinear mountain waves propagating in a moist atmosphere. This Lagrangian description is used to deduce an integral representation of the atmospheric pressure distribution in terms of the temperature within the laminar flow layer. Kirchoff’s equation is used to determine a temperature dependent enthalpy which together with the Clausius-Clapeyron equation is used to obtain an explicit expression for temperature and vapour pressure profiles in a saturated atmosphere where mountain waves are prominent. Precipitation rates are computed from the first law of thermodynamics and compare favourably with meteorological field data at Feldberg, a mountain in Germany. The second law of thermodynamics is used to show that there is a subregion near the tropopause at which precipitation is prohibited within the laminar flow.

Spectral stability analysis of ”anomalous” solitons and multi-solitons is presented in the context of a generalized Hamiltonian system called the Kaup-Kupershmidt (KK) equation. The KK equation is a completely integrable fifth order Korteweg-de Vries equation, which admits third order eigenvalue problem in its Lax pair. We also prove Hamiltonian-Krein index identities in verifying stability criterion of its multi-solitons. However, the KK equation does not possess the (L^2) conservation law and the linearized operators around the multi-solitons have no spectral gap. The main ingredients of the proof are new operator identities for second variation operator and completeness in (L^2) of the squared eigenfunctions of the third order eigenvalue problem for the KK equation. The operator identities and completeness relation are shown by employing the recursion operators of the KK equation.

In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in (L^1cap L^p) for (p>3/2). The aim of this note is to remove the constraint on p, showing that pathwise uniqueness holds for all (L^1cap L^p) initial data with arbitrary (p>1).

We consider some new estimates for general steady Navier–Stokes solutions in plane domains. According to our main result, if the domain is convex, then the difference between mean values of the velocity over two concentric circles is bounded (up to a constant factor) by the square-root of the Dirichlet integral in the annulus between the circles. The constant factor in this inequality is universal and does not depend on the ratio of the circle radii. Several applications of these formulas are discussed.