Journal of Mathematical Fluid Mechanics最新文献

We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling wave solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitability of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden & Raynaud] for solitary waves.

In this paper, we present a regularity criterion to the Navier–Stokes equations based on one entry of the velocity gradient. In fact, we prove that the weak solution to the Navier-Stokes equations is regular provided that (partial _{3}u_{3}in L^{beta }(0, T; L^{alpha }(mathbb {R} ^{3}))) with (alpha >frac{7+sqrt{13}}{6}) and:

where ( widehat{alpha }=frac{1}{alpha }.) This result improves the previous result obtained by Zujin Zhang and Yali Zhang in (Z. Angew. Math. Phys.)(2021), which states the similar result for (alpha ge frac{3+sqrt{17}}{4}.) Notice that (frac{7+sqrt{13}}{6}<frac{3+sqrt{17}}{4},) thus the range of (alpha ) is changed. Also, we show that (beta ) corresponding to (alpha ) which is obtained in our result is smaller than (beta ) obtained in the mentioned paper.

Finite-time blowup of solutions (u(x, t), b(x, t)) to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data ((u_0(x),b_0(x))) is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of (u_x) along Lagrangian trajectories that originate at a point (x_0) at which (b_0(x)) and (b_0'(x)) vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing (b_0'(x_0)) suppresses finite-time blowup.

The flow of two macroscopically immiscible, viscous, incompressible fluids with unmatched densities is studied, where a transfer of mass between the constituents by phase transition is taken into account. To this end, two quasi-incompressible diffuse interface models with singular free energies are analyzed, differing primarily in their velocity averaging. Firstly, to generalize a model by Abels, Garcke, and Grün, a thermodynamically consistent system of Navier–Stokes/Cahn–Hilliard type with source terms is derived in a framework of continuum fluid dynamics, followed by a proof of existence of weak solutions to the latter. Secondly, the quasi-stationary version of a model by Aki, Dreyer, Giesselmann, and Kraus is investigated analytically, with existence of weak solutions being established for the resulting quasi-stationary Stokes system coupled to a Cahn–Hilliard equation with a source term.

We prove the non-linear stability of a class of travelling-wave solutions to the dissipative Aw-Rascle system with a singular offset function, which is formally equivalent to the compressible pressureless Navier-Stokes system with a singular viscosity. These solutions encode the effect of congestion by connecting a congested left state to an uncongested right state, and may also be viewed as approximations of solutions to the ‘hard-congestion model’. By using carefully weighted energy estimates we are able to prove the non-linear stability of viscous shock waves to the Aw-Rascle system under a small zero integral perturbation, which in particular extends previous results that do not handle the case where the viscosity is singular.

The kinetic behavior of the physical entropy of viscous and heat-conductive fluids is an important and challenging problem, since the entropy equation possesses high degeneracy and singularity in the vacuum region. This article presents a conclusion that for the heat-conductive compressible nematic liquid system, the strong solution to the Cauchy problem is uniformly bounded in entropy and the (L^{2}) regularities of the velocity and temperature can be preserved, provided the initial density vanishes in the far field is less than (O left( frac{1}{|x |^{2}}right) ), which improved the previous work [18]. The proof relies on the singular weighted energy method and a modified De Giorgi type iterative technique.

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for 1D compressible Euler equations with damping of time and space dependent coefficient (alpha (x,t)), which models the compressible flow through porous media. We prove that the solutions to this system globally exist and converge to the diffusion waves, which are the self-similar solutions to the corresponding nonlinear parabolic equation given by Darcy’s law. The optimal convergence rates are also obtained. The proof is accomplished by virtue of energy estimates.

In this paper, we study the zero-viscosity limit of the two-dimensional MHD equations in the mixed Prandtl-Shercliff regime. Under the assumption that the initial tangential magnetic field is a non-zero constant, we justify the zero-viscosity limit for Sobolev initial data and obtain the optimal convergence rate in ({L}^{infty }) space.

We investigate the zero-Mach limit of compressible Navier-Stokes equations in 3D bounded domains with non-slip boundary condition. No smallness restrictions are imposed on the initial velocity and the time interval. If the limiting system admits a reasonably smooth solution on a certain period, we verify that the corresponding compressible system admits the smooth solution on the same duration as well, provided the Mach number is small enough. Moreover, the solutions of compressible system converge uniformly to that of the incompressible one as Mach number tends to zero. We apply the global geometric tools introduced by Chrisrodoulou-Lindblad [8] to get higher order estimates of the density near the boundary, which also help us relax the smallness condition (Vert nabla ^2rho _0Vert le Cvarepsilon ) in previous works to (Vert nabla ^2rho _0Vert le Cvarepsilon ^{-alpha }) for some (alpha ge 0).

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in Hölder-Zygmund spaces (C^r(mathbb {R}^2)) for (r>1) and uniformly local Sobolev spaces (H_{ul}^s(mathbb {R}^2)) for (s>2).