Global Existence Of Smooth Solutions Research Articles

On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium

AbstractIn this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e3,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the globalL1‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.

Structural stability of a 1D compressible viscoelastic fluid model

This paper is concerned with a compressible viscoelastic fluid model proposed by Öttinger. Although the model has a convex entropy, the Hessian matrix of the entropy does not symmetrize the system of first-order partial differential equations due to the non-conservative terms in the constitutive equation. We show that the corresponding 1D model is symmetrizable hyperbolic and dissipative and satisfies the Kawashima condition. Based on these, we prove the global existence of smooth solutions near equilibrium and justify the compatibility of the model with the Navier–Stokes equations.

Global existence of smooth solutions to 2D Chaplygin gases on curved space

This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two‐dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.

Global existence for small data of the viscous Green–Naghdi type equations

We consider the Cauchy problem for the Green–Naghdi equations with viscosity, for small initial data. It is well-known that adding a second order dissipative term to a hyperbolic system leads to the existence of global smooth solutions, once the hyperbolic system is symmetrizable and the so-called Kawashima–Shizuta condition is satisfied. In a previous work, we have proved that the Green–Naghdi equations can be written in a symmetric form, using the associated Hamiltonian. This system being dispersive, in the sense that it involves third order derivatives, the symmetric form is based on symmetric differential operators. In this paper, we use this structure for an appropriate change of variable to prove that adding viscosity effects through a second order term leads to global existence of smooth solutions, for small data. We also deduce that constant solutions are asymptotically stable.

Global smooth solutions of MHD equations with large data

In this paper, we establish the global existence of smooth solutions of the three-dimensional MHD system for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in the critical norm.

Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

In this paper, we investigate the smooth spherically symmetric solutions to 3D relativistic Chaplygin gases with variable entropy, whose initial data is assumed to be a small smooth perturbation to a constant state. Due to the structure of the equation, we can still take advantage of the “null condition” which is satisfied by the potential equation for isentropic and spacetime irrotational relativistic Chaplygin gases and obtain the global existence of smooth solution by continuity method. This generalizes the result for classical Chaplygin gases of Godin [J Math Pures Appl 87(9):91–117, 2007] to the relativistic case.

Global existence of smooth solutions to relativistic string equations in Schwarzschild space-time for small initial data

This paper is concerned with the motion of relativistic strings in Schwarzschild space-time. In a general framework, we first analyze the basic equations for the motion of a [Formula: see text]-dimensional extended object in a general enveloping space-time [Formula: see text], which is a given Lorentzian manifold, and then we investigate some important properties enjoyed by the equations for the motion of relativistic strings in Schwarzschild space-time. In particular, the equations are shown to form a totally linearly degenerate system of first-order hyperbolic equations in [Formula: see text] dimensions. Based on this observation and under suitable assumptions, we are able to prove the global existence of smooth solutions to the Cauchy problem for the equations of motion of relativistic strings (in Schwarzschild space-time) with sufficiently small arc length.

Remarks on nonlinear elastic waves in the radial symmetry in 2-D

In this paper, we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind of structural conditionon the nonlinearity in order to stop the formation of finite time singularitiesof local smooth solutions. In the radial symmetric case, inspired by Alinhac's work on 2-D quasilinear wave equations [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618], we show that such null condition can ensure the global existence of smooth solutions with small initial data.

No blow-up to a variational wave equation in liquid crystals

We establish the global existence of smooth solutions to the Cauchy problem for a system of variational wave equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and bend coefficients.

Existence and Asymptotic Behavior of Solutions to a Semilinear Hyperbolic-Parabolic Model of Chemotaxis

We consider a general hyperbolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts and using detailed decay estimates of the Green function.

Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.

Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations

Under the conditions of both the initial data being the small perturbation of given steady state solution and the boundary strength being small, the global existence of smooth solution to the initial boundary value problem of the relativistic Euler-Poisson equations is proved. The convergence of the global smooth solution to smooth steady state solution in time exponentially is also obtained when time goes to infinity.

Global existence and asymptotic behavior of the solutions to the 3D bipolar non-isentropic Euler–Poisson equation

In this paper, the global existence of smooth solutions for the three-dimensional (3D) non-isentropic bipolar hydrodynamic model is showed when the initial data are close to a constant state. This system takes the form of non-isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. Moreover, the L2-decay rate of the solutions is also obtained. Our approach is based on detailed analysis of the Green function of the linearized system and elaborate energy estimates. To our knowledge, it is the first result about the existence and L2-decay rate of global smooth solutions to the multi-dimensional non-isentropic bipolar hydrodynamic model.

Global Weak Solutions of 3D Compressible Nematic Liquid Crystal Flows with Discontinuous Initial Data and Vacuum

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic nematic liquid crystal flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L2, and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in L?, L2 and H1, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.

Some stability results on global solutions to the Navier–Stokes equations

Abstract In these notes we present some results concerning the existence of global smooth solutions to the three-dimensional Navier–Stokes equations set in the whole space. We are particularly interested in the stability of the set of initial data giving rise to a global smooth solution.

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

AbstractFor the physical vacuum free boundary problem with the sound speed being C1/2‐Hölder continuous near vacuum boundaries of the one‐dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self‐similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher‐order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher‐order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc.

Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier–Stokes equations

In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C1/2-Hölder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced.

Uniformly Global Smooth Solutions and Convergence of Euler--Poisson Systems with Small Parameters

We consider an Euler--Poisson system with small parameters arising in the modeling of unmagnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove the uniformly global existence of smooth solutions with respect to the parameters. This result allows us to show the global-in-time convergence of the Euler--Poisson system as each of the parameters goes to zero. The proof is based on unified energy estimates which are valid for all the parameters. The smallness conditions on the initial data are given explicitly in terms of the parameters.

Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to $L^1$ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal $L^2$ decay rates for the global smooth solutions and their spatial derivatives. At last, we establish the weak-strong uniqueness property in the class of finite energy weak solutions for the incompressible viscoelastic system.

Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum

In this paper, we study the global existence of weak solutions to the Cauchy problem for three-dimensional equations of compressible micropolar fluids with discontinuous initial data. Here it is assumed that the initial energy is suitably small in L-2, that the initial density is bounded in L-infinity, and the gradients of initial velocity and microrotational velocity are bounded in L-2. Particularly, this implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a by product, we also prove the global existence of smooth solutions with strictly positive density and small initial-energy.