Journal of Mathematical Fluid Mechanics最新文献

In this paper, we consider the singular points of suitable weak solutions to the 3D co-rotational Beris–Edwards system modeling the hydrodynamical motion of nematic liquid crystal flows, which is a coupled system with the Navier–Stokes equations for the fluid and a parabolic system of Q-tensor for the liquid average orientation. We prove that if ((textbf{u},Q)) defined on (mathbb {R}^{3}times (0,T)) is a suitable weak solution to the 3D co-rotational Beris–Edwards system, and satisfies

then for a given open subset (Omega subseteq mathbb {R}^{3}) and for a given moment of time (t_0in (0,T)), the number of points of the set (Sigma (t_0)cap Omega ) is finite, where (Sigma (t_0)equiv {(x,t_0)in Sigma }) and (Sigma ) is the set of singular points for ((textbf{u},Q)). Moreover, if (T_{1}in (0,T)) is the first time for singularity appears, and if ((textbf{u},Q)) satisfies

with (3<ple infty ) and (c_0) is a postive constant, then we show that ((textbf{u},Q)) preserves the energy equality on the closed interval ([0,T_{1}]) including the first blow-up time (T_{1}).

By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in (B^s_{p,r}(mathbb {R}^d)) with (s>max {1+frac{d}{2},frac{3}{2}}) and ((p,r)in (1,infty )times [1,infty )). This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of (B^s_{p,r}(mathbb {R}^d)) near the origin.

For two-dimensional steady pure-gravity water waves with a unidirectional flow of constant favorable vorticity, we prove an explicit bound on the amplitude of the wave, which decays to zero as the vorticity tends to infinity. Notably, our result holds true for arbitrary water waves, that is, we do not have to restrict ourselves to periodic or solitary or symmetric waves.

In this paper, employing the duality technique, we prove that the very weak solution of Magneto-Hydrodynamics equations is regular in (mathbb {R}^3times (0, T]) if it belongs to the Banach space (L^{p}(h,T;L^{q}(mathbb {R}^{3}))) with ( frac{2}{p}+frac{3}{q}=1, qin (3,infty )) for any small (h>0). Secondly, we further prove the integrability condition imposed on the magnetic field can be removed by using the energy method and the regularity theory of the heat operator, which is of independent interest.

The present paper is the continuation of works (Xu and Chi in Nonlinearity 34:164–204, 2021, Xu in The maximal regularity and its application to a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects in (L^{p})-type framework. arXiv:2201.05960, 2022). Under the assumption on the some large initial data, we obtain the existence of global strong solutions to a non-conservative viscous compressible two-fluid model with capillarity effects in any dimension (Nge 2). Our analysis mainly relies on Fourier frequency localization technology, commutator estimate and Bony’s decomposition.

In this survey, we study the boundary value problem for the stationary Navier–Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes stationary Navier–Stokes flows around cylindrical obstacles. Leray’s invading domains method is presented as a starting point. Then we discuss the boundedness and convergence of general D-solutions (solutions with finite Dirichlet integrals) in exterior domains. For the Leray solutions of the flow around an obstacle problem, we study the nontriviality, and the justification of the limit velocity at small Reynolds numbers. Further, under the same assumption of small Reynolds numbers the global uniqueness theorem for the problem is established in the class of D-solutions, its proof deals with the accurate perturbative analysis based on the linear Oseen system, inspired by classical Finn-Smith technique; the classical Amick and Gilbarg–Weinberger papers are involved here as well. The forced Navier–Stokes system in the whole plane is also presented as a closely related problem. A list of unsolved problems is given at the end of the paper.

A compressible, viscous and heat conducting fluid is confined between two parallel plates maintained at a constant temperature and subject to a strong stratification due to the gravitational force. We consider the asymptotic limit, where the Mach number and the Froude number are of the same order proportional to a small parameter. We show the limit problem can be identified with Majda’s model of layered “stack-of-pancake” flow.

In this paper, we study the initial-boundary value problem of the Landau–Lifshitz–Bloch equation in three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by the Maxwell equation, and the exchange field contains exchange constant. Firstly, we establish the existence of weak solutions of the equation by using the Faedo–Galerkin approximation. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends towards zero under appropriate compactness conditions. Moreover, we obtain an equation that can better describe the magnetic dynamic behavior of ferromagnetic films with negligible thickness at high temperatures.

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier–Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space (mathbb {R}^3), with initial value (vec u_0in textrm{BMO}^{-1}) (as in Koch and Tataru’s theorem) and with force (vec f={{,textrm{div},}}mathbb {F}) where smallness of (mathbb {F}) ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin’s solutions (in (L^2mathcal {F}^{-1}L^1)) or solutions in the multiplier space (mathcal {M}(dot{H}^{1/2,1}_{t,x}mapsto L^2_{t,x})).