Well-posedness in critical spaces for incompressible viscoelastic fluid system

In this paper, we study the well-posedness in critical spaces of incompressible viscoelastic fluid system of Oldroyd model. Precisely, for this system with initial data in Besov space of critical regularity, we prove the existence and uniqueness of the local solution, which is also shown to exist globally in time provided the initial data is small under certain norms.     

Global existence in critical spaces for compressible Navier-Stokes equations

We investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium. We obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations. More precisely, the initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations, and one more derivative is needed for the density. We point out a smoothing effect on the velocity and a L ¹-decay on the difference between the density and the constant reference state. The proof lies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term. Oblatum 9-II-1999 & 6-I-2000?Published online: 29 March 2000     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

On the sharp singular limit for slightly compressible fluids

We consider the equations of motion to slightly compressible fluids and we prove that solutions converge, in the strong norm, to the solution of the equations of motion of incompressible fluids, as the Mach number goes to zero. From a physical point of view this means the following. Assume that we are dealing with a well-specified fluid, so slightly compressible that we assume it to be incompressible. Our result means that the distance between the (continuous) trajectories of the real and of the idealized solution is ‘small’ with respect to the natural metric, i.e. the metric that endows the data space.     

Global well-posedness of the non-isentropic full compressible magnetohydrodynamic equations

In this paper, we are concerned with Cauchy problem for the multi-dimensional (N ≥ 3) non-isentropic full compressible magnetohydrodynamic equations. We prove the existence and uniqueness of a global strong solution to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our method is mainly based on the uniform estimates in Besov spaces for the proper linearized system with convective terms.     

Global existence and asymptotic dynamics in a 3D fractional chemotaxis system with singular sensitivity

In this paper, we investigate the global existence and asymptotic dynamics of solutions to a fractional singular chemotaxis system in three dimensional whole space. We deal with the new difficulties arising from fractional diffusion by using Riesz transform and Kato-Ponce’s commutator estimates appropriately, and establish the local existence of solution. Then with the help of combining the local existence and the a priori estimates, the global existence and uniqueness of solution with small initial data is derived. Moreover, we obtain the asymptotic decay rates of solution by the method of energy estimates.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

On the existence and uniqueness of smooth solution for a generalized Zakharov equation

The paper deals with the existence and uniqueness of smooth solution for a generalized Zakharov equation. We establish local in time existence and uniqueness in the case of dimension d=2,3. Moreover, by using the conservation laws and Brezis-Gallouet inequality, the solution can be extended globally in time in two dimensional case for small initial data. Besides, we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data.     

Global existence of classical solution for a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum

In this work, we obtain the global existence and uniqueness of classical solutions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowed. We get the upper and lower bounds of gas and liquid masses n and m by the continuity methods which we use to study the compressible Navier-Stokes equations.     

Global solution to the incompressible flow of liquid crystals

The initial–boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.     

From the dynamics of gaseous stars to the incompressible Euler equations

A model for the dynamics of gaseous stars is introduced and formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus Tn is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Local solvability of equations of magnetohydrodynamics on the whole space

Using the technique of expanding domains, we prove the existence of a weak, local in time solution to the equations of magnetohydrodynamics, derived from the equations for viscous, compressible and heat-conducting fluids, on the whole space under special assumptions on pressure and entropy. Compared with the same approach for barotropic compressible fluids, we show how to overcome loss of the global integrability of temperature and velocity fields in corresponding spaces.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.     

On second grade fluids with vanishing viscosity

We consider in ℝⁿ (n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.     

Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows

An initial-boundary value problem is considered for the density-dependent incompressible viscous magnetohydrodynamic flow in a three-dimensional bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. For the initial density away from vacuum, the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.     

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined, which turns out to be a Besov space. Similarly to 3D-Navier Stokes, the uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.     

On the incompressible limit of inviscid compressible fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. Here we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

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Sunto Nel presente lavoro consideriamo le equazioni di Eulero per un fluido inviscido comprimibile barotropico in un dominio limitato. è ben noto che, quando il numero di Mach tende a zero, i moti comprimibili approssimano le soluzioni delle equazioni del moto relative ad un fluido inviscido incomprimibile. Qui discutiamo, per il problema al contorno, i diversi tipi di convergenza sotto differenti ipotesi sui dati, in particolare la convergenza debole nel caso di dati iniziali uniformemente limitati e la convergenza forte nella norma dello spazio dei dati.

     

Global Existence in Critical Spaces for Density-Dependent Incompressible Viscoelastic Fluids

In this paper we consider the local and global well-posedness to the density-dependent incompressible viscoelastic fluids. We first study some linear models associated to the incompressible viscoelastic system. Then we approximate the system by a sequence of ordinary differential equations, by means of the Friedrichs method. Some uniform estimates for those solutions will be obtained. Using compactness arguments, we will get the local existence up to extracting a subsequence by means of Ascoli’s lemma. With the help of small data conditions and hybird Besov spaces, we finally derive the global existence.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.