Journal of Mathematical Fluid Mechanics最新文献

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.

The present paper studies a two-component mathematical model representing shallow-water wave propagation primarily in equatorial ocean regions, incorporating the effects of weak Coriolis force and equatorial undercurrent. We start with the Green–Naghdi type equations under the weak Coriolis and equatorial undercurrent effects from the Euler equations, then the two-component Camassa–Holm system with the two effects is derived by truncating asymptotic expansions of the quantities to the appropriate order. Analytically, we study the mathematical properties of the solutions to the two-component Camassa–Holm system including the ill-posedness of the solutions in Besov spaces (B^{s}_{p,infty }times B^{s-1}_{p,infty }) with (1le ple infty ) and (s>max left{ 2+frac{1}{p},frac{5}{2}right} ), the Hölder continuity of the data-to-solution map in Besov spaces (B^{s}_{p,r}times B^{s-1}_{p,r}) with (1le p,rle infty ) and (s>max left{ 2+frac{1}{p},frac{5}{2}right} ). We then investigate the Gevrey regularity and analyticity of the system in ({G_{delta ,s}^{gamma }}times {G_{delta ,s-1}^{gamma }}) with (delta ge 1, nu>gamma >0) and (s>frac{5}{2}). Finally, we provide the persistence properties and the spatial asymptotic profiles of the solutions in weighted spaces (L ^ p_{phi }=L^p(mathbb {R},phi ^pdx)).

We prove two Liouville-type theorems for the stationary Navier–Stokes equations in (mathbb {R}^3) under some assumptions on 1) the growth of the (L^s) mean oscillation of a potential function of the velocity field, or 2) the relative decay of the head pressure and the square of the velocity field at infinity. The main idea is to use Saint-Venant type estimates to characterize the growth of Dirichlet energy of nontrivial solutions. These assumptions are weaker than those previously known of a similar nature.

We consider the construction of linear instability of parallel shear flows, which was developed by Lin (SIAM J Math Anal 35(2):318–356, 2003). We give an alternative simple proof in Sobolev setting of the problem, which exposes the mathematical role of the Plemelj–Sochocki formula in the emergence of the instability, as well as does not require the cone condition. Moreover, we localize this approach to obtain an approximation of the Kelvin–Helmholtz instability of a flat vortex sheet.

We construct in the paper the low-regularity strong solutions to the viscous surface wave equations in anisotropic Sobolev spaces. By using the Lagrangian structure of the system to homogenize the free boundary conditions coupled with the semigroup method of the linear operator, we establish a new iteration scheme on a known equilibrium domain to get the low-regularity strong solutions, in which no compatibility conditions of the accelerated velocity on the initial data are required.