Journal of Mathematical Fluid Mechanics最新文献

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.

In this paper, we perform the fast rotation limit (varepsilon rightarrow 0^+) of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip (Omega _varepsilon :=,{mathbb {R}}^2times , left. right] -ell _varepsilon ,ell _varepsilon left[ right. ,), where (varepsilon in ,left. right] 0,1left. right] ) is the size of the Rossby number and (ell _varepsilon >0) for any (varepsilon >0). By letting (ell _varepsilon longrightarrow 0^+) for (varepsilon rightarrow 0^+) and considering Navier-slip boundary conditions at the boundary of (Omega _varepsilon ), we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a 3-D setting. In this respect, we remark that the case (ell _varepsilon geqslant ell >0) for all (varepsilon >0) remains largely open at present.

We investigate compressible nonisothermal diffuse interface fluid models also termed capillary fluids. Such fluid models involve van der Waals’ gradient energy, Korteweg’s tensor, Dunn and Serrin’s heat flux as well as diffusive fluxes. The density gradient is added as an extra variable and the convective and capillary fluxes of the augmented system are identified by using the Legendre transform of entropy. The augmented system of equations is recast into a normal form with symmetric hyperbolic first order terms, symmetric dissipative second order terms and antisymmetric capillary second order terms. New a priori estimates are obtained for such augmented system of equations in normal form. The time derivatives of the parabolic components are less regular than for standard hyperbolic–parabolic systems and the strongly coupling antisymmetric fluxes yields new majorizing terms. Using the augmented system in normal form and the a priori estimates, local existence of strong solutions is established in an Hilbertian framework.

A mechanical interaction of compressible viscoelastic fluids with viscoelastic solids in Kelvin–Voigt rheology using the concept of higher-order (so-called 2nd-grade multipolar) viscosity is investigated in a quasistatic variant. The no-slip contact between fluid and solid is considered and the Eulerian-frame return-mapping technique is used for both the fluid and the solid models, which allows for a “monolithic” formulation of this fluid–structure interaction problem. Existence and a certain regularity of weak solutions is proved by a Schauder fixed-point argument combined with a suitable regularization.