Journal of Mathematical Fluid Mechanics最新文献

In this paper, we establish the existence of the global weak admissible solution for the Cauchy problem of a N-peakon system in the sense of (H^1(mathbb {R})) space under a sign condition. Second, we claim that the global weak admissible solution for the system with the same initial data is not unique by giving a example. Finally, an image of the solutions of the above example which does not satisfy the uniqueness is given, which makes it easier to see the properties of non-uniqueness more intuitively.

This paper is concerned with a three-dimensional Keller–Segel–Navier–Stokes system incorporating singular flux limitation and superlinear production. The primary goal is to establish global existence of weak solutions under conditions ensuring that flux limitations suppress the blow-up tendencies induced by superlinear growth. More precisely, this paper focuses on the system

in a bounded domain (Omega subset mathbb {R}^3) with smooth boundary, where (0< alpha < 1) and (beta ge 1). Under the assumption (alpha > 1 - frac{1}{3beta -1}), we prove global existence of weak solutions to the Neumann problem for ((*)). This study extends the previous work by Winkler [27], in which the corresponding system with the regular sensitivity ((|nabla c|^2+1)^{-frac{alpha }{2}}) and the linear production ((beta =1)) was considered, and highlights how strong flux limitation can control the effects of superlinear growth.

We consider the Cauchy problem to the three-dimensional isentropic compressible Magnetohydrodynamics (MHD) system with density-dependent viscosities. When the initial density is linearly equivalent to a large constant state, we prove that strong solutions exist globally in time, and there is no restriction on the size of the initial velocity and initial magnetic field. As far as we know, this is the first result on the global well-posedness of density-dependent viscosities with large initial data for 3D compressible MHD equations.

In the current state of the art regarding the Navier–Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins’ inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a (C^1) velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure Appl. Math 72(7), 1351–1385 (2019), we conclude the uniqueness of the solution.

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.