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1.
In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L2 norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.  相似文献   

2.
We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein‐Maxwell‐scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space‐time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space‐time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.  相似文献   

3.
The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with H1‐initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the L2‐norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. Copyright © 2017 John Wiley & Sons, Ltd.  相似文献   

4.
We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L2 initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L2 initial data. © 2001 John Wiley & Sons, Inc.  相似文献   

5.
In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H2 initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L‐norm in terms of the Lq‐norms up to a logarithm of the Lp‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H2 estimates for global regularity.© 2016 Wiley Periodicals, Inc.  相似文献   

6.
We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ? u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L3 ∕ 2‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

7.
We prove the existence of a global strong solution in some class of Marcinkiewicz spaces for the micropolar fluid in an exterior domain of R3, with initial conditions being a non‐smooth disturbance of a steady solution. We also analyse the large time behaviour of those solutions and apply our results in the context of the Navier–Stokes equations. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

8.
We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

9.
We propose and analyze an application of a fully discrete C2 spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H1 norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L2 and H1 norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.  相似文献   

10.
We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The Lr‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L1 and L decay rates of its first order derivatives with respect to space variables, are derived by using Lq ? Lr estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

11.
We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ0, u 0 satisfy the compatibility condition for some radially symmetric g ∈ L2. The initial density ρ0 needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.  相似文献   

12.
We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L1‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L1‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L1‐spaces. For soft potentials, in L1‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.  相似文献   

13.
This paper addresses strong cosmic censorship for spacetimes with self‐gravitating collisionless matter, evolving from surface‐symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature k of the symmetric surfaces and the cosmological constant Λ. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if Λ=0 or k≤0, Λ > 0. In the case Λ > 0, k=1, we give a detailed geometric characterization of possible “boundary” components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild–de Sitter‐type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all k,Λ. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from i + with strictly positive limiting area radius.© 2016 Wiley Periodicals, Inc.  相似文献   

14.
We consider a compressible viscous fluid with the velocity at infinity equal to a strictly non‐zero constant vector in ?3. Under the assumptions on the smallness of the external force and velocity at infinity, Novotny–Padula (Math. Ann. 1997; 308 :439– 489) proved the existence and uniqueness of steady flow in the class of functions possessing some pointwise decay. In this paper, we study stability of the steady flow with respect to the initial disturbance. We proved that if H3‐norm of the initial disturbance is small enough, then the solution to the non‐stationary problem exists uniquely and globally in time, which satisfies a uniform estimate on prescribed velocity at infinity and converges to the steady flow in Lq‐norm for any number q? 2. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

15.
An iteration scheme is used to show the well‐posedness of the initial‐boundary value problem for incompressible hypoelastic materials, which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume that the stress is a rank‐one matrix T=qqT, q∈?n, and develop energy estimates to show that the problem is locally well‐posed. This problem is related to incompressible ideal magnetohydrodynamics (MHD). We show that the general case T=CCT, C∈?n×n can be handled by a generalization of the method we developed. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

16.
We consider the long‐time behavior and optimal decay rates of global strong solution to three‐dimensional isentropic compressible Navier–Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space with l?4 and s∈[0, 1], we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates (1 + t)?3/4?s/2 in the L2‐norm or (1 + t)?3/2?s/2 in the L‐norm, respectively, which are shown to be optimal for the CNS system. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

17.
We consider the third‐order Claerbout‐type wide‐angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There is strong indication that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well‐posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second‐order boundary condition, derived by assuming that the standard (narrow‐angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial‐boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second‐order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge‐type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson–Kreiss boundary condition in the wide‐angle case. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

18.
We propose a result of local existence and uniqueness of a mild solution to the one‐dimensional Vlasov–Poisson system. We establish the result for an initial condition lying in the space W1,1(?2), then we extend it to initial conditions lying in the space BV(?2), without any assumption of continuity, boundedness or compact support. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

19.
We establish the existence of local in time semi‐strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three‐dimensional domains with boundary uniformly of class C3. Under suitable assumptions, uniqueness of local semi‐strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

20.
In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi‐strong solutions and get some higher order estimates, and then prove the global well‐posedness of the two‐dimensional Boussinesq system with variable viscosity for H3 initial data. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

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