Journal of Mathematical Fluid Mechanics最新文献

We discuss optimal estimates of solutions to the compressible Navier–Stokes equations in Besov norms. In particular, we consider the estimate of the curl-free part of the solution to the linearised equations, in the homogeneous case. We prove that our estimate is optimal in the (L^infty )-norm by showing that the norm is bounded from below by the same decay rate.

We study the Boussinesq approximation for the incompressible Euler equations using Lagrangian description. The conditions for the Lagrangian fluid map are derived in this setting, and a general method is presented to find exact fluid flows in both the two-dimensional and the three-dimensional case. There is a vast amount of solutions obtainable with this method and we can only showcase a handful of interesting examples here, including a Gerstner type solution to the two-dimensional Euler–Boussinesq equations. In two earlier papers we used the same method to find exact Lagrangian solutions to the homogeneous Euler equations, and this paper serves as an example of how these same ideas can be extended to provide solutions also to related, more involved models.

We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.

We consider a model of one dimensional isentropic compressible Navier–Stokes equations for which the classical Newtonian flow is replaced by a Maxwell flow. We establish the asymptotic stability of rarefaction waves for this model under some small conditions on initial perturbations and amplitude of the waves. The proof is based on (L^2) energy methods.

We consider a diffuse interface model for an incompressible binary fluid flow. The model consists of the Navier–Stokes–Voigt equations coupled with the mass-conserving Allen–Cahn equation with Flory–Huggins potential. The resulting system is subject to generalized Navier boundary conditions for the (volume averaged) fluid velocity ({{textbf {u}}}) and to a dynamic contact line boundary condition for the order parameter (phi ). These boundary conditions account for the moving contact line phenomenon. We establish the existence of a global weak solution which satisfies an energy inequality. A similar result is proven for the Allen–Cahn–Navier–Stokes system. In order to obtain some higher-order regularity (w.r.t. time) we propose the Voigt approximation: in this way we are able to prove the validity of the energy identity and of the strict separation property. Thanks to this property, we can show the uniqueness of quasi-strong solutions, even in dimension three. Regularization in finite time of weak solutions is also shown.

The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler–Lagrange equations we proceed to derive some model equations for different propagation regimes. While the long-wave regime reproduces the well known KdV equation, the short- and intermediate long wave regimes lead to highly nonlinear and nonlocal evolution equations.

The question of whether the two-dimensional inviscid Boussinesq equations can develop a finite-time singularity from general initial data is a challenging open problem. In this paper, we obtain two new regularity criteria for the local-in-time smooth solution to the two-dimensional inviscid Boussinesq equations. Similar result is also valid for the nonlocal perturbation of the two-dimensional incompressible Euler equations.

The combustion model is studied in three-dimensional (3D) smooth bounded domains with various types of boundary conditions. The global existence and uniqueness of strong solutions are obtained under the smallness of the gradient of initial velocity in some precise sense. Using the energy method with the estimates of boundary integrals, we obtain the a priori bounds of the density and velocity field. Finally, we establish the blowup criterion for the 3D combustion system.

This paper is concerned with the two-dimensional incompressible Euler-like equations. More precisely, we consider the system with only damping in the vertical component equation. When the domain is the whole space (mathbb {R}^2), it is well known that solutions of the incompressible Euler equations can grow rapidly in time while solutions of the Euler equations with full damping are stable. As the intermediate case of the two equations, the global well-posedness and the stability in (mathbb {R}^2) remain the outstanding open problem. Our attentions here focus on the domain (Omega =mathbb {T}times mathbb {R}) with (mathbb {T}) being 1D periodic box. Compared with (mathbb {R}^2), the domain (Omega ) allows us to separate the physical quantity f into its horizontal average (overline{f}) and the corresponding oscillation (widetilde{f}). By deriving the strong Poincaré inequality and two anisotropic inequalities related to (widetilde{f}), we are able to employ the time-weighted energy estimate to establish the stability of the solution and the precise large-time behavior of the system provided that the initial data is small and satisfies the reflection symmetry condition.

We propose a new concept of weak solution to the equations of compressible magnetohydrodynamics driven by ihomogeneous boundary data. The system of the underlying field equations is solvable globally in time in the out of equilibrium regime characteristic for turbulence. The weak solutions comply with the weak–strong uniqueness principle; they coincide with the classical solution of the problem as long as the latter exists. The choice of constitutive relations is motivated by applications in stellar magnetoconvection.