Journal of Mathematical Fluid Mechanics最新文献

It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations with Dirichlet boundary conditions if the viscosity (nu ) is small enough, provided the horizontal wave number (alpha ) lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this article, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.

In this paper, we present an analysis of the Kelvin-Helmholtz instability in two-dimensional ideal compressible elastic flows, providing a rigorous confirmation that weak elasticity has a destabilizing effect on the Kelvin-Helmholtz instability. There are two critical velocities, (U_{text {low}}) and (U_{text {upp}}), where (U_{text {low}}) and (U_{text {upp}}) represent the lower and upper critical velocities, respectively. We demonstrate that when the magnitude of the rectilinear solutions satisfies (U_{text {low}}+cepsilon _{0}le |dot{v}^{+}_{1}| le U_{text {upp}}-cepsilon _{0}), the linear and nonlinear ill-posedness of the piecewise smooth solutions of the Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic flows is established uniformly with respect to the background velocity in the interval ([U_{text {low}}+cepsilon _{0}, U_{text {upp}}-cepsilon _{0}]), where c is the sound speed and (epsilon _{0}) is some small enough positive constant.

In this paper, we investigate a nonstationary Oseen-Navier-Stokes system for incompressible fluids under leak and slip boundary conditions, along with state constraints. The weak form of the system leads to a class of evolutionary quasi-variational hemivariational inequalities with nonhomogeneous initial conditions. By demonstrating the weak compactness of the solution set to the inequality problem, we establish the weak compactness of the solution set for the constrained nonstationary incompressible Oseen-Navier-Stokes system with mixed boundary value conditions. We then examine an optimal control problem associated with the system, which is motivated by important applications such as artificial heart models. General existence and compactness results for the optimal control problem are established.

We consider axisymmetric, swirl-free solutions of the Euler equations in three and higher dimensions, of generalized anti-parallel-vortex-tube-pair-type: the initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity. We prove lower bounds on the growth of such solutions in all dimensions. In particular in three dimensions, we improve a recent lower bound of Choi and Jeong [5].

This paper studies the global existence of large perturbation solutions for the thin film growth model, a class of fourth-order nonlinear parabolic equations. The main purpose of this paper is to attempt to find an effective method to capture the enhanced dissipation mechanism generated by the fourth-order parabolic equation and the Couette flow in the full-space case through the Green’s function, thereby suppressing the blow-up of the solutions for the nonlinear parabolic equation and obtaining the overall existence of the solutions.

Based on the nonlinear ocean vorticity equation derived by Constantin and Johnson (see [9]), this paper derives a vorticity equation incorporating variable eddy viscosity through the selection of appropriate parameters and suitable asymptotic approximations. The exact solution of the equatorial vorticity equation for the Pacific between (160^{circ })E and (80^{circ })W is provided, while a set of cubic functions is employed to describe the easterly jet stream above the thermocline (z=-T) in close proximity to the surface of the Equator, as well as the westerly strong jet stream near (z=0). Moreover, we obtain the expression of the corresponding pressure field.

Analytical solutions to Fluid-Structure Interaction (FSI) problems are almost absent in the literature. However, they are crucial for validation and convergence analysis of numerical methods, as well as for providing insight into the complex coupling dynamics between fluids and solids. In this paper, we derive two analytical and one semi-analytical solutions for three FSI problems, spanning a class of solutions by varying their geometrical and physical parameters. All solutions exhibit complex nonlinear behaviours, which we compare with numerical simulations using a monolithic method. These three FSI problems are described in the cylindrical coordinates, drawing inspiration from Couette flow, with two of them featuring a moving fluid-solid interface and the third incorporating a nonlinear constitutive solid model. To the best of our knowledge, for the first time, we present FSI problems with analytical solutions that include a moving interface.

This paper investigates the long-wave asymptotic behavior of three-dimensional Euler-Poisson system describing cold ion plasmas, both in the unmagnetized case and in the case with a uniform magnetic field. Through an appropriate scaling which balances the nonlinearity and dispersion, we derive two decoupled Kadomtsev-Petviashvili (KP)/Zakharov-Kuznetsov (ZK) equations from the original system. A rigorous justification of the long-wave approximation is given by establishing uniform estimates of the difference between the solutions of Euler-Poisson system and a suitable constructed approximation profile. It demonstrates that solutions of Euler-Poisson system in unmagnetic case are well approximated by the two-way waves from the corresponding KP-II type equations, while the solutions of the system with magnetic field are convergent to the counter directional waves of the ZK equations.

This article is devoted to the mathematical study of a new Navier-Stokes-alpha model with a nonlinear filter equation. For a given indicator function, this filter equation was first considered by W. Layton, G. Rebholz, and C. Trenchea in [Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow, J. Math. Fluid Mech. 14: 325-354 (2012)] to select eddies for damping based on the understanding of how nonlinearity acts in real flow problems. Numerically, this nonlinear filter equation was applied to the nonlinear term in the Navier-Stokes equations to provide a precise analysis of numerical diffusion and error estimates. Mathematically, the resulting alpha-model is described by a doubly nonlinear parabolic-elliptic coupled system. We therefore undertake the first theoretical study of this system by considering periodic boundary conditions in the spatial variable. Specifically, we address the existence and uniqueness of weak Leray-type solutions, their rigorous convergence to weak Leray solutions of the classical Navier-Stokes equations, and their long-time dynamics through the concept of the global attractor and some upper bounds for its fractal dimension. Handling the nonlinear filter equation together with the well-known nonlinear transport term makes certain estimates delicate, particularly when deriving upper bounds on the fractal dimension. For the latter, we adapt techniques developed for hyperbolic-type equations.

We consider the incompressible Hall-MHD system on the 3D whole space. Then, it is known that the perturbed system around the constant equilibrium state exhibits a dispersive structure. However, this dispersion is so complicated that the results on the effect of dispersion are known only for the special case (nu =mu ), where the dispersive relation becomes simpler. Here (nu ) and (mu ) are viscosity and resistive coefficients respectively. The purpose of this paper is to improve the previous results and investigate the dispersive effect for the general case (nu ne mu ) without complicated calculations. Consequently, we may obtain the global well-posedness and time-periodic solvability for large data in critical Besov spaces, provided that the size of the constant magnetic field is sufficiently large.