We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible \emph{heat-conducting} viscoelastic rate-type fluids with stress-diffusion, subject to a stick-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the extra stress tensor. In the introductory section we develop the thermodynamic foundations of the proposed model, and we document the role of thermodynamics in obtaining critical structural relations between the quantities of interest. These structural relations are then exploited in the mathematical analysis of the governing equations. In particular, the definition of weak solution is motivated by the thermodynamic basis of the model.
The extra stress tensor describing the elastic response of the fluid is in our case purely spherical, which is a simplification from the physical point of view. The model nevertheless exhibits features that require novel mathematical ideas in order to deal with the technically complex structure of the associated internal energy and the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of large-data global-in-time weak solutions to the governing equations for \emph{coupled thermo-mechanical processes} in viscoelastic rate-type fluids.
The initial value problem for Hookean incompressible viscoelastictic motion in three space dimensions has global strong solutions with small displacements.
The global existence of solutions to incompressible viscoelastic flows has been a longstanding open problem, even for the global weak solution. Under some special structure ("div-curl" condition) the global small smooth solution was obtained in \cite{llzhou, cz}. However, the result with large initial data remains unknown up to now, and it is studied in this paper. We shall put forward a new structure: the cone-condition and then derive the global smooth solution to 3D incompressible viscoelastic system with a class of large initial data. The key is to gain an angle quantity in the estimate of…
In this paper we analyze a 2D free-boundary viscoelastic fluid model of Oldroyd-B type at infinite Weissenberg number. Our main goal is to show the existence of the so-called splash singularities, namely points where the boundary remains smooth but self-intersects. The combination of existence and stability results allows us to construct a special class of initial data, which evolve in time into self-intersecting configurations. To this purpose we apply the classical conformal mapping method and later we move to the Lagrangian framework, as a consequence we deduce the existence of splash singu…
This paper studies the inviscid limit of the two-dimensional incompressible viscoelasticity, which is a system coupling a Navier-Stokes equation with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known since the work of F. H. Lin, C. Liu, and P. Zhang in "On hydrodynamics of viscoelastic fluids". The inviscid case was solved recently by the second author Z. Lei. in "Global well-posedness of incompressible elastodynamics in two dimensions". While the latter was solely based on the techniques fro…
This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the globa…
In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by constructing a Lyapunov functional. Moreover, it is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.
We analyze a mathematical model that describes the interaction between an insulating rigid body and an incompressible electrically conducting fluid surrounding it. The model as well as the mathematical analysis involve the fields of fluid-structure interaction and of magnetohydrodynamics. Our main result shows the existence of a weak solution to the corresponding system of partial differential equations. The proof relies on a use of a time discretization via the Rothe method to decouple the system. This allows us to deal with test functions, depending on the position of the moving body and the…
In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $Ω$ in $\mathbb{R}^d\,\, (d\geq 2)$. We prove that under the Shinbrot type conditions $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } {\bf F} \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $, the boundary conditions $u|_{\part…
Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls
We consider a model for an incompressible visoelastic fluid. It consists of the Navier-Stokes equations involving an elastic term in the stress tensor and a transport equation for the evolution of the deformation gradient. The novel feature of the paper is the introduction of the notion of a dissipative solution and its analysis. We show that dissipative solutions exist globally in time for arbitrary finite energy initial data and that a dissipative solution and a strong solution emanating from the same initial data coincide as long as the latter exists.
Strong well-posedness of a fluid--poro-viscoelastic interaction problem: An approach by Spectral analysis
Princeton University（普林斯顿大学）；Technische Universität Darmstadt（达姆施塔特工业大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯科，巴斯克科学基金会）
We investigate finite amplitude stability of spatially inhomogeneous steady state of an incompressible viscoelastic fluid which occupies a mechanically isolated vessel with walls kept at spatially non-uniform temperature. For a wide class of incompressible viscoelastic models including the Oldroyd-B model, the Giesekus model, the FENE-P model, the Johnson--Segalman model, and the Phan--Thien--Tanner model we prove that the steady state is stable subject to any finite perturbation.
In this paper, we prove the global existence of small smooth solutions to the three-dimensional incompressible Oldroyd-B model without damping on the stress tensor. The main difficulty is the lack of full dissipation in stress tensor. To overcome it, we construct some time-weighted energies based on the special coupled structure of system. Such type energies show the partial dissipation of stress tensor and the strongly full dissipation of velocity. In the view of treating "nonlinear term" as a "linear term", we also apply this result to 3D incompressible viscoelastic system with Hookean elast…
In this work, we introduce a notion of dissipative weak solution for a system describing the evolution of a heat-conducting incompressible non-Newtonian fluid. This concept of solution is based on the balance of entropy instead of the balance of energy and has the advantage that it admits a weak-strong uniqueness principle, justifying the proposed formulation. We provide a proof of existence of solutions based on finite element approximations, thus obtaining the first convergence result of a numerical scheme for the full evolutionary system including temperature dependent coefficients and visc…
We consider the system of partial differential equations governing two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion, involving a general objective derivative. The studied system generalizes the incompressible Navier--Stokes equations for the fluid velocity $v$ and pressure $p$ by the presence of an additional term in the constitutive equation for the Cauchy stress expressed in terms of a positive definite tensor $B$. The tensor $B$ evolves according to a diffusive variant of an equation that can be viewed as a combination of corresponding counterpa…
On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type
Charles University, Faculty of Mathematics and Physics（查理大学数学与物理学院）；University of Warsaw, Faculty of Mathematics, Informatics and Mechanics（华沙大学数学、信息学和机械学院）；Institute of Mathematics of Polish Academy of Sciences（波兰科学院数学研究所）
In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. First, we prove that the model under consideration posseses a global strong solution in a suitable functional framework. Second, we derive stability estimates with respect to an external magnetic field. Based on the stability estimates we use the …
In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ with a moving flat boundary $\{Y=0\}$. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the s…
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…
We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density, nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic $α$-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field f…
We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity $v$, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor $\mathbb{B}$. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of p…
