Aaron Brunk, Yong Lu, Maria Lukacova-Medvidova
In this paper we analyze the three-dimensional Peterlin viscoelastic model. By means of a mixed Galerkin and semigroup approach we prove the existence of a weak solutions. Further combining parabolic regularity with the relative energy method we derive a conditional weak-strong uniqueness result.
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We prove the existence of weak solutions to a viscoelastic phase separation problem in two space dimensions. The mathematical model consists of a Cahn-Hilliard-type equation for two-phase flows and the Peterlin-Navier-Stokes equations for viscoelastic fluids. We focus on the case of a polynomial-like potential and suitably bounded coefficient functions. Using the Lagrange-Galerkin finite element method complex behavior of solution for spinodal decomposition including transient polymeric network structures is demonstrated.
The aim of this paper is to prove global in time existence of weak solutions for a viscoelastic phase separation. We consider the case with singular potentials and degenerate mobilities. Our model couples the diffusive interface model with the Peterlin-Navier-Stokes equations for viscoelastic fluids. To obtain the global in time existence of weak solutions we consider appropriate approximations by solutions of the viscoelastic phase separation with a regular potential and build on the corresponding energy and entropy estimates.
We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation gradient and for the magnetization obtained from a special case of the micromagnetic energy. It turns out that the conditions on uniqueness coincide with those for the well-known Navier-Stokes equations in bounded domains: weak solutions are unique in two spatial dimensions, and weak solutions satisfying the Prodi-Serrin conditions are unique among all weak solut…
The existence and uniqueness of the global strong solution with small initial data to the three-dimensional viscoelastic fluids is established.
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 aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the non-convex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model we present the estimates in three space dimensions and derive conditional relative energy estimates.
Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks
本文研究时间相关裂纹区域中的双曲型偏积分微分系统，给出保证解的唯一性及其对裂纹连续依赖性的条件。
The aim of this work is to prove the global-in-time existence of weak solutions for a viscoelastic phase separation model in three space dimensions. To this end we apply the relative energy concept provided by [3]. We consider the case of regular polynomial-type potentials and positive mobilities, as well as the degenerate case with logarithmic potential and vanishing mobility.
Existence and weak-strong uniqueness for damage systems in viscoelasticity
本文针对粘弹性材料损伤的偏微分方程系统，证明了弱解的全局存在性和强解的局部存在性，并利用相对能量不等式建立了强解在弱解类中的唯一性。
We study the uniqueness and regularity of the steady states of the diffusively driven Boltzmann equation in the physically relevant case where the restitution coefficient depends on the impact velocity including, in particular, the case of viscoelastic hard-spheres. We adopt a strategy which is novel in several aspects, in particular, the study of regularity does not requires a priori knowledge of the time-dependent problem. Furthermore, the uniqueness result is obtained in the small thermalization regime by studying the so-called quasi-elastic limit for the problem. An important new aspect li…
A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of $L^2$-based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit est…
Uniqueness of dynamic elastography for isotropic standard linear solid viscoelastic media
本文基于麦拉洪和尤恩的论证，证明了在各向同性扩展麦克斯韦模型和各向同性扩展标准线性固体模型中，通过单次测量可唯一确定感兴趣区域内的剪切波速。
We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored energies satisfying an Andrews-Ball condition, allowing for non convexity in a compact set. Existence of weak solutions with deformation gradients in $H^1$ is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored en…
A new model for viscoelastic phase separation is proposed, based on a systematically derived conservative two-fluid model. Dissipative effects are included by phenomenological viscoelastic terms. By construction, the model is consistent with the second law of thermodynamics, and we study well-posedness of the model, i.e., existence of weak solutions, a weak-strong uniqueness principle, and stability with respect to perturbations, which are proven by means of relative energy estimates. A good qualitative agreement with mesoscopic simulations is observed in numerical tests.
Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows
本文研究了$\mathbb{R}^d$中不可压缩粘弹性流体的Cauchy问题，通过Helmholtz投影的新分解证明了全局光滑解的存在性，获得了最优$L^2$衰减率，并建立了有限能量弱解的弱强唯一性。
The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represen…
We consider the flows of viscoelastic fluid which obey a constitutive law of integral type. The existence and uniqueness results for solutions of the initial boundary value problem are proved, and the stationary case is studied.
We consider a family of linear viscoelastic shells with thickness $2\varepsilon$ ( $\varepsilon$ , small parameter), clamped along a portion of their lateral face, all having the same middle surface $S$. We formulate the three-dimensional mechanical problem in curvilinear coordinates and provide existence and uniqueness of (weak) solution of the corresponding three-dimensional variational problem. We are interested in studying the limit behavior of the three-dimensional problems and their solutions (displacements $\mathbf{u}^\varepsilon$ of components $u_i^\varepsilon$) when $\varepsilon$ tend…
Blow-Up Criteria and Weak--Strong Uniqueness for Compressible Fluid--Viscoelastic Shell Interactions
本文研究了压缩流体-粘弹性壳体相互作用系统强解的有限时间爆破准则及弱-强唯一性原理，通过高阶能量估计和Lipschitz型控制保证解的延展性。
We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of implicit constitutive relations we establish the existence and uniqueness of a global-in-time large-data weak solution. We then focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain ten…
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.
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.
In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal i…
In this paper we introduce a system coupling a nonlinear Schrödinger equation with a system of viscoelasticity, modeling the interaction between short and long waves, acting for instance on media like plasmas or polymers. We prove the existence and uniqueness of local (in time) strong solutions and the existence of global weak solutions for the corresponding Cauchy problem. In particular we extend previous results in [Nohel \emph{et. al.}, Commun. Part. Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish with some numerical computations to illustrate our results.
A dynamic model for viscoelastic materials with prescribed growing cracks
该研究针对含时变裂纹域、粘度系数可能退化的粘弹性动力系统，证明了解的存在性与更强正则性假设下的唯一性，并通过例子说明裂纹扩展会产生额外能量耗散。
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…
The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceas…
In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a glo…
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…
Weak and mild solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces
本文研究三维 MHD 方程与带阻尼粘弹性 Navier–Stokes 方程，在 Wiener amalgam 空间中证明温和解存在性并构造全局局部能量弱解，同时推广局部能量解的正则性与小–大唯一性结论。
In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of for the displacement and the parabolic hemivariational inequality for the temperature. The existence of s…
Weak solutions and weak-strong uniqueness for a compressible power-law-Oldroyd--B fluid model
研究了在 $R^{3}$ 中具有剪切增稠特性的粘弹性可压缩幂律-Oldroyd--B流体模型，证明了在增长指数大于等于 $\ rac 52$ 时弱解的存在性，并在附加应力张量初始值满足额外可积性条件下确立了强弱唯一性。
The configurational distribution function, solution of an evolution (diffusion) equation of the Fokker-Planck-Smoluchowski type, is (at least part of) the corner stone of polymer dynamics: it is the key to calculating the stress tensor components. This can be reckoned from \cite{bird2}, where a wealth of calculation details is presented regarding various polymer chain models and their ability to accurately predict viscoelastic flows. One of the simplest polymer chain idealization is the Bird and Warner's model of finitely extensible nonlinear elastic (FENE) chains. In this work we offer a proo…
We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+ηu(1,t)=g(t), u(x,0)=û_{0}(x), u_{t}(x,0)={û}_{1}(x), where η\geq 0, q\geq 2 are given constants {û}_{0}, {û}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under …
We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this pa…
Local Well-Posedness For Barotropic Compressible Fluid-Viscoelastic Shell Interactions
该研究采用完全欧拉框架与局部Hanzawa变换，证明了三维正压可压缩Navier-Stokes流与粘弹性壳相互作用强解的局部时间存在唯一性，补充了相关弱理论并拓展了强边界结果的适用范围。
Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
本研究针对黏弹塑性流体模型，证明了强解的短时间存在性及在Leray--Hopf型弱解中的唯一性，并在应力扩散消失时引入能量变分解概念，通过相对能量不等式取极限证明其存在性。
Analysis of nonlinear poroviscoelastic flows with discontinuous porosities
本文研究孔隙度不连续的多孔介质非线性孔隙黏弹性流动，通过孔隙度与有效压力的非线性相互作用建模，证明了解的存在唯一性，并给出跳跃不连续初始孔隙度下问题适定的条件。
We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characteristic field is linearly degenerate. We show the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.
The one-dimensional fractional derivative Maxwell model (e.g. Palade et al. Rheol. Acta 35, 265, 1996), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade et al. Int. J. Non-Linear Mech. 37, 315, 1999, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade J. Math. Phys. 49, 043101, 2008, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve t…
Global Smooth Solutions to a Thermoelastic Cauchy Problem in Phase Transitions
本文研究了一维粘弹性相变问题，通过Ginzburg-Landau能量模型和非凸三次应力-应变定律，证明了Cauchy问题的全局光滑解存在性和唯一性，结合了行波分解、指数变换和耦合能量估计。
Weak and strong solutions for polymeric fluid-structure interaction of Oldroyd-B type
本研究针对描述二维稀聚合物流体与一维粘弹性壳相互作用的Oldroyd-B哑铃模型，证明了弱解存在性、唯一强解的全局存在性（结构无退化前提下）及无条件弱-强唯一性，允许流体采用任意非退化参考域。
Vanishing center-of-mass limit of the 2D-1D corotational Oldroyd-B polymeric fluid-structure interaction problem
研究了具有质心扩散的2D共旋转Oldroyd-B聚合物流体与1D粘弹性壳体相互作用系统，证明了当质心扩散系数趋于零时强解收敛至无扩散系统的弱解，并得出弱-强唯一性结论。
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