In this paper, we prove the global existence of general small solutions to compressible viscoelastic system. We remove the "initial state" assumption ($\tilde ρ_0 \det F_0 =1$) and the "div-curl" structure assumption compared with previous works. It then broadens the class of solutions to a great extent, more precisely the initial density state would not be constant necessarily, and no more structure is need for global well-posedness. It's quite different from the elasticity system in which structure plays an important role. Since we can not obtain any dissipation information for density and deformation tensor, we introduce a new effective flux in the thought of regarding the wildest "nonlinear term" as "linear term". Although the norms of solution may increase now, we can still derive the global existence for it.
The formation of singularity and breakdown of classical solutions to the three-dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite-time formation of singularity in classical solutions is proved for certain initial data. For the compressible viscoelastic fluids, a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.
The existence and uniqueness of the local strong solution to the three-dimensional compressible viscoelastic fluids near the equilibrium is established. In addition to the uniform estimates on the velocity, some essential uniform estimates on the density and the deformation gradient are also obtained.
This work is devoted to the study of a compressible viscoelastic fluids satisfying the Oldroyd-B model in a regular bounded domain. We prove the local existence of solutions and uniqueness of flows by a classical fixed point argument.
The existence and uniqueness of local in time strong solution with large initial data for the three-dimensional compressible viscoelastic fluid is established. The strong solution has weaker regularity than the classical solution. The Lax-Milgram theorem and the Schauder-Tychonoff fixed-point argument are applied.
The global existence of strong solution to the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids near equilibrium is established in a bounded domain. Uniform estimates in $W^{1,q}$ with $q>3$ on the density and deformation gradient are also obtained. All the results apply to the two-dimensional case.
We consider the system of equations describing motion of compressible viscoelastic fluids in a whole space. We investigate the large time behavior of solutions around a motionless state, and obtain the $L^p$ decay estimates of solutions for $1<p\leq\infty$, provided that the initial data is sufficiently close to the motionless state. In addition, we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave.
We develop a mathematical theory for a class of compressible viscoelastic rate-type fluids with stress diffusion. Our approach is based on the concepts used in the nowadays standard theory of compressible Newtonian fluids as renormalization, effective viscous flux identity, compensated compactness. The presence of the extra stress, however, requires substantial modification of these techniques, in particular, a new version of the effective viscous flux identity is derived. With help of these tools, we show the existence of global--in--time weak solutions for any finite energy initial data.
We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate regularities of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of flow map so that the inherent structure of the elastic te…
We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial preturbation belongs to $W^{2,1}$, and is sufficiently small in $H^4\cap L^1$, the solutions grow in time at the same rate as $t^{\frac{1}{2}}$ in $L^1$ due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.
In this paper, we are concerned with the convergence rates of the global strong solution to constant equilibrium state for the compressible viscoelastic fluids in the whole space. We combine both analysis about Green's matrix method and energy estimate method to get optimal time decay rate in critical Besov space framework. Our result imply the optimal $L^{2}$-time decay rate and only need the initial data to be small in critical Besov space which have very low regularity compared with traditional Sobolev space.
The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic-parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the {scaling} of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the $L^1-$decay of the density and deformation gradient.
The inviscid limit for the two-dimensional compressible viscoelastic equations on the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions …
We consider the large time behavior of global strong solutions to the compressible viscoelastic flows on the whole space $\mathbb{R}^N\,(N\geq 2)$, where the system describes the elastic properties of the compressible fluid. Adding a suitable initial condition involving only the low-frequency, we prove optimal time decay estimates for the global solutions in the $L^p$ critical regularity framework, which are similar to those of the compressible Navier-Stokes equations. Our results rely on the pure energy argument, which allows us to remove the usual smallness assumption of the data in the low-…
In this paper, we are concerned with the global existence and optimal rates of strong solutions for three-dimensional compressible viscoelastic flows. We prove the global existence of the strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. If additionally the initial data belong to $L^1$, the optimal convergence rates of the solutions in $L^p$-norm with $2\leq p\leq 6$ and optimal convergence rates of their spatial derivatives in $L^2$-norm are obtained.
We are interested in the multi-dimentional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. The new ingredient lies in the introduction of effective flux $(θ,\mathcal{G})$, which enable…
This paper is dedicated to the global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows in the whole space $\mathbb{R}^n$ with any $n\geq2$. We aim at extending those works by Qian \& Zhang and Hu \& Wang to the critical $L^p$ Besov space, which is not related to the usual energy space. With aid of intrinsic properties of viscoelastic fluids as in \cite{QZ1}, we consider a more complicated hyperbolic-parabolic system than usual Navier-Stokes equations. We define "\emph{two effective velocities}", which allows us to cancel out the coupling among th…
Semilinear parabolic systems with bi-linear nonlinearities cover a lot of applications and their optimal control leads to relatively simple optimality conditions. An example is the incompressible Navier-Stokes system for homogeneous fluids, which is however here modified towards a physically reasonable model of slightly (so-called ``semi'') compressible liquids rather than fully compressible gases. An optimal control problem optimizing also pressure on the boundary is considered and, in the simple variant, analysed as far as uniqueness of the control-to-state mapping and 1st-order optimality c…
We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in o…
We investigate local and global strong solutions for the incompressible viscoelastic system of Oldroyd--B type. 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 Navier--Stokes equations, and one more derivative is needed for the deformation tensor. We point out a smoothing effect on the velocity and a $L^1-$decay on the difference between the deformation tensor and the identity matrix. Our result implies that the…
We investigate global strong solutions for the incompressible viscoelastic system of Oldroyd--B type with the initial data close to a stable equilibrium. We obtain the existence and uniqueness of the global solution in a functional setting invariant by the scaling of the associated equations, where the initial velocity has the same critical regularity index as for the incompressible Navier--Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible Navier-Stokes, one may construct the unique global solution for a class of large highly osci…
In this paper we first employ the energetic variational method to derive a micro-macro model for compressible polymeric fluids. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We then prove the global in time existence of the smooth solution near the global equilibrium.
The Cauchy problem of the compressible Oldroyd-B model without damping mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in density and stress tensor in the model is compensated by exploiting an intrinsic structure and introducing new quantities between density, velocity and stress tensor. Therefore, global solutions to the system with small initial data in critical Besov spaces are obtained. As a byproduct, optimal time decay rates of the solutions are derived by using an energy estimation argument. The results remain valid for the compressible viscoelastic system without …
A mechanical interaction of compressible viscoelastic fluids with viscoelastic solids in Kelvin-Voigt rheology using the concept of higher-order (so-called 2nd-grade multipolar) viscosity is investigated in a quasistatic variant. The no-slip contact between fluid and solid is considered and the Eulerian-frame return-mapping technique is used for both the fluid and the solid models, which allows for a "monolithic" formulation of this fluid-structure interaction problem. Existence and a certain regularity of weak solutions is proved by a Schauder fixed-point argument combined with a suitable reg…
In this paper, we mainly study the global strong solutions and its long time decay rates of all order spatial derivatives to a micro-macro model for compressible polymeric fluids with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first prove that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for $d\geq2$. Moreover, for $d\geq3$, we also show a new critical Fourier estimation that allow us to give the long time decay rates of $L^2$…
On the relaxation towards mechanical equilibrium for two-pressure compressible flows
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；IMJ-PRG（数学研究所巴黎分部）；Université Paul Sabatier（图卢兹第三大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université Claude Bernard Lyon 1（里昂第一大学）；Institut Camille Jordan（卡米尔·若尔当研究所）
In this paper, we mainly study the large time behavior to a 2D micro-macro model for compressible polymeric fluids with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. Firstly the Fourier splitting method yields that the logarithmic decay rate. By virtue of the time weighted energy estimate, we can improve the decay rate to $(1 + t)^{-\frac{1}{4}}$. Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov spaces with negative index and obtain the…
We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. The result of this paper mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preven…
Bidimensional incompressible viscous flows with well-localised vorticity are well-known to develop vortex structures. The purpose of the present paper is to recover the asymptotic profiles describing these phenomena for homogeneous finite-energy flows as asymptotic profiles for near-equilibrium isentropic compressible flows. This task is performed by extending the sharp description of the asymptotic behaviour of near-equilibrium compressible flows obtained by David Hoff and Kevin Zumbrun to the case of finite-energy vortex-like solutions.
A kinetic-fluid model describing the evolutions of disperse two-phase flows is considered. The model consists of the Vlasov-Fokker-Planck equation for the particles (disperse phase) coupled with the compressible Navier-Stokes equations for the fluid (fluid phase) through the friction force. The friction force depends on the density, which is different from many previous studies on kinetic-fluid models and is more physical in modeling but significantly more difficult in analysis. New approach and techniques are introduced to deal with the strong coupling of the fluid and the particles. The glob…
We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the $L^2$-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.
The system of compressible adiabatic flow through porous media is considered in $\mathbb{R}^{3}$ in the present paper. The global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium. We also show that the pressure of the system converges to its equilibrium state at the same $L^2$-rate $(1+t)^{-3/4}$ as the Navier-Stokes equations without heat conductivity, but the velocity of the system decays at the $L^2$-rate $(1+t)^{-5/4}$, which is faster than the $L^2$-rate $(1+t)^{-3/4}$ for the Navier-Stokes equations without heat conductivity [3].
This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit de…
We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial data. The results hold in both two and three dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals and mixture problems.
Compressible Navier-Stokes-Landau-Lifshitz-Gilbert system: derivations and well-posedness
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Finance and Mathematics, Huainan Normal University（淮南师范学院金融与数学学院）；School of Mathematics, Hunan University（湖南大学数学学院）；School of Mathematics and Physics, China University of Geosciences（中国地质大学数理学院）
In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier-Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the fi…
Recently, for periodic initial data with initial density allowed to vanish, Huang and Li [1] establish the global existence of strong and weak solutions for the two-dimensional compressible Navier{Stokes equations with no restrictions on the size of initial data provided the bulk viscosity coefficient is λ= ρ^βwith β> 4/3. Moreover, the large-time behavior of the strong and weak solutions are also obtained, in which the velocity gradient strongly converges to zero in L^2 norm. In this note, we further point out that the velocity strongly converges to an equilibrium velocity in H^1 norm, in whi…
Motivated by \cite{CG10,CZ6}, we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.
In recent paper 5, it is shown that the upper decay rate of global solution of compressible Navier-Stokes(CNS) equations converging to constant equilibrium state $(1, 0)$ in $H^1-$norm is $(1+t)^{\frac34(\frac{2}{p}-1)}$ when the initial data is large and belongs to $H^2(\mathbb{R}^3) \cap L^p(\mathbb{R}^3) (p\in[1,2))$. Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in $H^1-$norm is $(1+t)^{-\frac32(\frac1p-\frac12)-\frac12}$. For the case of $p=1$, the lower bound of decay rate for the global solut…
