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1.
In this paper, we establish a blowup criterion for the two-dimensional compressible nematic liquid crystal flows. The criterion is given in terms of the density and the gradient of direction field, where the later satisfies the Serrin-type blowup criterion. For this result, we do not need the initial density to be positive.  相似文献   

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We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.  相似文献   

5.
We consider smooth three-dimensional spherically symmetric Eulerian flows of ideal polytropic gases outside an impermeable sphere, with initial data equal to the sum of a constant flow with zero velocity and a smooth perturbation with compact support. Under a natural assumption on the form of the perturbation, we obtain precise information on the asymptotic behavior of the lifespan as the size of the perturbation tends to 0. When there is no sphere, so that the flow is defined in all space, corresponding results have been obtained in [P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions, Arch. Ration. Mech. Anal. 177 (2005) 479–511].  相似文献   

6.
In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902 and the Doctorial Foundation of National Educational Ministry 20050246001.  相似文献   

7.
This paper concerns subsonic flows passing a two-dimensional duct for the steady compressible Euler system. If the Bernoulli constant is uniform in the flow field, the density at the entry and both the pressures at the entrance and the exit are given, we show that the problem is generally ill-posed; but if we give the pressure at the exit with a constant difference, then under the same other conditions as above we establish the existence of subsonic flows.  相似文献   

8.
In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.  相似文献   

9.
We prove that the uniform stability at permanently acting disturbances of a given solution of the Navier-Stokes equations for viscous compressible isothermic fluid is a consequence of the uniform exponential stability of the zero solution of so-called linearized equations.The research was supported by the grant No. 201/93/2177 of Grant Agency of Czech Republic.  相似文献   

10.
In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).  相似文献   

11.
The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.  相似文献   

12.
We show some new uniqueness results for compressible flows with data having critical regularity. In the barotropic case, uniqueness is stated whenever the space dimension N satisfies N ≥ 2, and in the polytropic case, whenever N ≥ 3. The endpoints N = 2 in the barotropic case and N = 3 in the polytropic case were left open in [4], [5] and [6].  相似文献   

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The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.  相似文献   

15.
We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R3. We first prove the local existence of solutions (ρ,u) in C([0,T*]; (ρ +H3(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T**)×Ω for some T** ∈ (0,T*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.  相似文献   

16.
In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument. Finally, the relation between reproductive and periodic in time (regular) solutions will be pointed out, differenting the 2D and 3D cases. Basically, in two-dimensional domains every reproductive solution is regular and time periodic, whereas the problem remains open for three-dimensional domains.  相似文献   

17.
We consider steady compressible Navier-Stokes-Fourier system for a gas with pressure p and internal energy e related by the constitutive law p=(γ−1)?e, γ>1. We show that for any there exists a variational entropy solution (i.e. solution satisfying the weak formulation of balance of mass and momentum, entropy inequality and global balance of total energy). This result includes the model for monoatomic gas (). If , these solutions also fulfill the weak formulation of the pointwise total energy balance.  相似文献   

18.
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω0×(0,L)∈R3. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L(0,L;L2(Ω0)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.  相似文献   

19.
We consider a free boundary problem for the equations of spherically symmetric motion of a isentropic gas with a density-dependent viscosity , where and λ are positive constants. We prove that the problem admits a weak solution provided that 0 < λ < 1/4.   相似文献   

20.
In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.  相似文献   

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