Journal of Mathematical Fluid Mechanics最新文献

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a ‘scaling gap’ between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework-based on a suitably defined ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field—in which the scaling gap between the regularity class and the corresponding a priori bound vanishes as the order of the derivative goes to infinity.

In this paper we study the compressible magnetohydrodynamics equations in three dimensions, which offer a good model for plasmas. Formation of singularity for (C^1)-solution in finite time is proved with axisymmetric initial data. The key observation is that the magnetic force term admits a good structure with axisymmetric assumption.

In this paper, we prove that the maximum norm of velocity divergence controls the breakdown of smooth (strong) solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier–Stokes equations with zero heat conduction. The results indicate that the nature of the blowup for the full compressible Navier–Stokes equations with zero heat conduction of viscous flow is similar to the barotropic compressible Navier–Stokes equations and does not depend on the temperature field. The main ingredient of the proof is a priori estimate to the pressure field instead of the temperature field and weighted energy estimates under the assumption that velocity divergence remains bounded. Furthermore, the initial vacuum states are allowed, and the viscosity coefficients are only restricted by the physical conditions.

We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to (H^{2+alpha ,1+alpha /2}), (alpha >5/8).

We consider the time-dependent Navier–Stokes system coupled with the heat equation governed by the nonlinear Tresca boundary conditions. We propose a discretization of these equations that combines Euler implicit scheme in time and finite element approximations in space. We present optimal error estimates for velocity, pressure and temperature. Numerical examples are displayed to illustrate the theoretical results.

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains (C^{1,alpha })-regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., (C^infty ) in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the (C^{1,alpha })-regularity assumption to be optimal. We also quantify the (C^infty )-regularity via high order energy estimates which, in particular, implies the well-posedness of the problem.

We consider in an infinite horizontal layer the stationary motion of a viscous compressible fluid in a magnetic field subject to the gravitational force, where the Dirichlet boundary condition for the velocity and similar but non-homogeneous and large enough conditions for the magnetic field are assumed. Existence of a stationary solution in a neighborhood close to the equilibrium state is obtained in Sobolev spaces as limit of a sequence of fixed points of some suitable operators.

In this paper we mainly study large time behavior for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. The sharp (L^2) decay rate was obtained on the co-rotational case. We prove that the optimal (L^2) decay rate of the velocity of the general FENE dumbbell model is ((1+t)^{-frac{d}{4}}) with (dge 2). Our obtained result is sharp and improves considerably the previous result in Luo and Yin (Arch Ration Mech Anal 224(1):209–231, 2017).

In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity (mu ) and the bulk one (lambda =brho ^beta ), here b is a positive constant, (beta ge 0). This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that (gamma >1), the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.

In this article, we establish the Wong–Zakai approximation result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection–diffusion equation, the Cahn–Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo–Galerkin approximation method, compactness arguments, and Prokhorov’s and Skorokhod’s representation theorems to ensure the existence of a probabilistically weak solution for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada–Watanabe theorem allows us to conclude the existence of a probabilistically strong solution (analytically weak solution) for the approximating system. Subsequently, we establish the Wong–Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong–Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work’s functional framework.