Journal of Mathematical Fluid Mechanics最新文献

In this paper, we investigate the long-time behaviors of the ideal compressible non-isentropic magnetohydrodynamic (MHD) equations, alternatively named the Lundquist equations with non-constant entropy. To be specific, we show that a finite-time breakdown of the ideal MHD system will occur eventually by deriving a differential inequality of blowup type in terms of a functional weighted by the radius of the spatial variable and given initial data only. Our result complements some existing result, in which the author considered the unweighted radial component of momentum. Moreover, our blowup result is independent of the initial magnetic field, as long as it has compact support, the magnetic permeability constant and the sign of the initial mass functional.

The problems of regularity and uniqueness are open for the supercritically dissipative surface quasi-geostrophic equations in certain classes. In this note we examine the extent to which small or large scales are necessarily active both for the temperature in a hypothetical blow-up scenario and for the error in hypothetical non-uniqueness scenarios, the latter understood within the class of Marchand’s solutions. This extends prior work for the 3D Navier-Stokes equations. The extension is complicated by the fact that mild solution techniques are unavailable for supercritical SQG. This forces us to develop a new approach using energy methods and Littlewood-Paley theory.

We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain (Omega ) with initial velocity (u_0) square-integrable, divergence-free and tangent to (partial Omega ). We supplement the equations with the Navier friction boundary conditions (u cdot {varvec{n}}= 0) and ([(2Su){varvec{n}}+ alpha u]_{{scriptstyle {textrm{tan}}}} = 0), where ({varvec{n}}) is the unit exterior normal to (partial Omega ), (Su = (Du + (Du)^t)/2), (alpha in C^0(partial Omega )) is the boundary friction coefficient and ([cdot ]_{{scriptstyle {textrm{tan}}}}) is the projection of its argument onto the tangent space of (partial Omega ). We prove global existence of a weak Leray-type solution to the resulting initial-boundary value problem and exponential decay in energy norm of these solutions when friction is positive. We also prove exponential decay if friction is non-negative and the domain is not a solid of revolution. These two results are well known in the case of Dirichlet boundary condition, but, even if they have been implicitly used for the Navier boundary conditions, the comprehensive analysis is not available in the literature. After carefully studying the Stokes semigroup for such a boundary condition, we use the Galerkin method for existence, Poincaré-type inequalities, with suitable adaptations to account for the differential geometry of the boundary, and a novel integral Gronwall-type inequality. In addition, in the frictionless case (alpha = 0), we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution.

The modified Camassa-Holm-Kadomtsev–Petviashvili equation is a two-dimensional extension of the modified Camassa-Holm equation. In this paper, we will demonstrate that the modified Camassa-Holm-Kadomtsev–Petviashvili equation allows for solitary wave solutions with a single peak, both on a line and on a circle.

In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space (mathbb {R}^d,;(din {2,3}). The two phases of the mixture are separated by a ({mathscr {C}}^{1+alpha })-regular sharp interface ({mathcal {C}}) across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is (alpha )-Hölder continuous on both sides of ({mathcal {C}}). The initial velocity belongs to the Sobolev space (H^1(mathbb {R}^d)), and the divergence of the initial stress tensor belongs to (L^2(mathbb {R}^d)). The later assumption expresses somehow the continuity of the normal component of the stress tensor. This result is more general than the one by Tani [Two-phase free boundary problem for compressible viscous fluid motion. Journal of Mathematics of Kyoto University 24(2): 243–267, 1984] as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.

We analyse a model for thermal convection in a class of generalized Navier-Stokes equations containing fourth order spatial derivatives of the velocity and of the temperature. The work generalises the isothermal model of A. Musesti. We derive critical Rayleigh and wavenumbers for the onset of convective fluid motion paying careful attention to the variation of coefficients of the highest derivatives. In addition to linear instability theory we include an analysis of fully nonlinear stability theory. The theory analysed possesses a bi-Laplacian term for the velocity field and also for the temperature field. It was pointed out by E. Fried and M. Gurtin that higher order terms represent micro-length effects and these phenomena are very important in flows in microfluidic situations. We introduce temperature into the theory via a Boussinesq approximation where the density of the body force term is allowed to depend upon temperature to account for buoyancy effects which arise due to expansion of the fluid when this is heated. We analyse a meaningful set of boundary conditions which are introduced by Fried and Gurtin as conditions of strong adherence, and these are crucial to understand the effect of the higher order derivatives upon convective motion in a microfluidic scenario where micro-length effects are paramount. The basic steady state is the one of zero velocity, but in contrast to the classical theory the temperature field is nonlinear in the vertical coordinate. This requires care especially dealing with nonlinear theory and also leads to some novel effects.

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.