Local and Global Solvability in a Viscous Wave Equation Involving General Temperature-Dependence

1 Implicit Functions.- 1.1 Formal solvability.- 1.2 Theorem on local solvability.- 1.3 Transformations of equations.- 1.4 Global solvability.- 1.5 Comments and references.- 2 Classification of One-dimensional Mappings.- 2.1 Wandering and non-wandering subsets.- 2.2 Mappings with wandering compact sets.- 2.2.1 Strictly monotonic mappings without fixed points.- 2.2.2 The Abel and cohomological equations.- 2.2.3 Smooth and analytic solutions of a cohomological equation.- 2.3 Local structure of mappings at an isolated fixed point.- 2.3.1 Formal classification.- 2.3.2 Smooth classification.- 2.3.3 Analytic classification.- 2.4 Diffeomorphisms with isolated fixed points.- 2.4.1 Topological classification.- 2.4.2 Smooth classification of diffeomorphisms with a unique fixed point.- 2.4.3 Smooth classification of diffeomorphisms with several hyperbolic fixed points.- 2.4.4 Another approach to smooth classification.- 2.5 One-dimensional flows and vector fields.- 2.5.1 Classification of vector fields in a neighborhood of a singular point.- 2.5.2 Flows on the real line with hyperbolic fixed points.- 2.6 Embedding problem and iterative roots.- 2.6.1 Mappings without non-wandering points.- 2.6.2 C0-embedding.- 2.6.3 Diffeomorphisms with a unique fixed point.- 2.6.4 Diffeomorphisms with several fixed points.- 2.7 Comments and references.- 3 Generalized Abel Equation.- 3.1 Local solvability.- 3.1.1 Local solvability in a neighborhood of a non-fixed point.- 3.1.2 Proof of Theorem 3.1 for analytic functions.- 3.1.3 Local solvability at an isolated fixed point.- 3.1.4 More on analytic solutions.- 3.2 Global solutions of equations with not more than one fixed point.- 3.2.1 Equations with fixed-point free mappings F.- 3.2.2 The case of a single fixed point.- 3.3 Gluing method for linear equations with several fixed points.- 3.3.1 Cohomological equation.- 3.3.2 Equations with hyperbolic fixed points.- 3.4 Comments and references.- 4 Equations with Several Transformations of Argument.- 4.1 Local solvability.- 4.2 Extension of solutions.- 4.2.1 Absorbers.- 4.2.2 Extension of solutions from an absorber.- 4.2.3 Extension from intersection of absorbers. Decomposition method.- 4.3 Examples.- 4.4 Difference equations in Carleman classes.- 4.4.1 Decomposition in classes C(mn).- 4.4.2 Equations with constant coefficients.- 4.4.3 Equations with non-constant coefficients.- 4.5 Comments and references.- 5 Linear Equations.- 5.1 Generalized linear Abel equation.- 5.1.1 Equations on the real line with a unique fixed point.- 5.1.2 Cohomological equation.- 5.1.3 Spectrum of a weighted shift operator.- 5.1.4 Normal solvability of equations with hyperbolic fixed points.- 5.1.5 Equations with periodic points.- 5.2 Localization of obstacles to solvability.- 5.3 Equations with constant coefficients.- 5.4 Equation with affine transformations of argument.- 5.5 Comments and references.

In 1980 BEALE [2] studied the problem of describing the motion of a layer of heavy, viscous, incompressible fluid lying above an infinite rigid bottom and having a non-compact free surface with no surface tension. He proved its local solvability in an anisotropic Sobolev space for any initial data. Its global solvability was discussed by SYLV~SXER [20]. But a crucial point of her proof does not appear clear to me: the regularity of the free surface. With surface tension taken into account, BEALE [31 and BEALE & NISHIDA [4] respectively studied the large-time existence and regularity and the large-time behavior of the solution to this problem with initial data near equilibrium. TERAMOTO [24, 25] considered these problems for fluids lying above an inclined plane. For any initial data the same problem was solved locally in time by ALLAIN [11 in the two-dimensional case. The aim of this paper is to establish the analogous result in the three-dimensional case. A similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by SOLONNIKOV. He proved local solvability in a H61der space in [12] and global solvability in the space W 2' 1, p > n (where n is the dimension of the domain) in [16] without surface tension and in [-13 15, 17 19,21] with surface tension. Let us formulate our problem. Given an initial domain f~ _ R 3 with x3 being the vertical component and an initial velocity vector field Vo in f~, we want to know the domain f~(t), t > 0, occupied by the fluid, which is bounded by the fixed bottom SB and the free surface Se(t), the velocity vector field v = v(x, t) = (vl, v2, v3) and the pressure p = p(x, t) so that

The initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type used for modeling nonstationary processes in semiconductors is examined. It is proved that this problem is uniquely solvable at least locally in time. Sufficient conditions for the problem to be solvable globally in time are found, as well as sufficient conditions for the local (but not global) solvability. In the case of only local solvability, upper and lower estimates for the time when a solution exists are determined in the form of either explicit or quadrature formulas.

Healthy articular cartilage is characterized by extremely low friction and high compressive stiffness. This dual-functionality is tailored by its biphasic structure, whereby a fluid phase interacts with the extracellular matrix. Osteoarthritis (OA) causes structural changes, thereby altering the biomechanical and frictional properties. How the structural and functional properties of human cartilage are associated with OA remain unknown. To address this, we identified relationships between structural parameters, viscoelastic and frictional properties of degenerated human cartilage through correlation analyses. We found that cartilage friction was mainly influenced by its microscopic structure, while the viscoelastic properties were also related to the macroscopic structure. The viscoelastic and frictional properties displayed a weak correlation. These findings provide insights into the interplay between cartilage structure and its functional properties in OA, which might provide a basis for advancements in diagnosing and treating degenerated human cartilage. Statement of significanceOsteoarthritis causes changes in the cartilages biphasic structure, thereby affecting functionality by altered biomechanical and frictional properties. Currently a cartilage-preserving therapeutic option remains lacking, because the disease is not fully understood. In our correlation analyses, we investigated relationships between the structural, the viscoelastic and frictional properties of degenerated human cartilage. We found that cartilage friction was particularly dependent on the microscopic structure, while the viscoelastic properties also correlated with the macroscopic structure. The frictional properties displayed only a weak dependency with the viscoelastic properties. These new insights into the structure-function and inter-functional relationships may provide new options to advance the diagnosis and treatment of degenerated cartilage.

The study of equations of mathematical physics, including inverse problems, is relevant today. This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse problem for a quasilinear pseudoparabolic equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas. The main types of the inverse problems are: boundary, retrospective, coefficient and geometric. The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems. Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems. In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution. Key words: Pseudoparabolic equation, inverse problem, existence, uniqueness, local solvability, global solvability, non-local condition.

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.

We study the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient. We obtain sufficient conditions for both global and local (in time) solvability. In the case of local (but not global) solvability, we obtain upper and lower bounds for the existence time of the solution in the form of explicit and quadrature formulas.

<p style='text-indent:20px;'>A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.</p>

We consider the solvability of non-invariant differential operators on homogeneous spaces. Such operators cannot be expected to have solutions in smooth functions (an illustrative example is provided). However, Lion has shown that, under suitable growth conditions on the infinitesimal components of the operators in a representation-theoretic decomposition, one can deduce solvability in a space of distributions. In this paper we prove that Lion's result can be improved to yield solvability in square-integrable functions. AN IMPROVEMENT OF LION'S RESULT Let G be a connected Lie group, H a closed subgroup, v a unitary representation of H and T = IndH Gv. Denote by it(g) the (complexified) universal enveloping algebra of the Lie algebra g of G. The elements of it(g) act on smooth sections in the space A of T by differential operators. The action is as follows. Let JH ,G denote the quotient of the modular functions of H and G. Let 'G ,H denote a relatively invariant measure on G/H. Consider C (T) = {f E C (G, ): f(gh) = 2 (h)v(h)lf(g), hEH,gEG}. The space t?? of C?-vectors of T lies inside C??(T). The action of G on L2 (G, , dl' ,H) by left translation is unitary, differentiates to g and extends canonically to it(g). The resulting action of it(g) on C??(T) gives the desired differential operators. For L E 11(g), we denote the corresponding differential operator on C??(T) by LT or T(L); and we denote the totality of them by iTA. We write itA for those that commute with the action of G, and refer to them as invariant. Typically, one has nice solvability results on the operators in tA.0 For example, if G/H is nilpotent and symmetric, and v = 1, we have global solvability [1]; or if G = G, x G,, H = AG the diagonal (v =1 again), then itA equals 3(go), the center of it(gl), and one has local solvability [3]. The main thrust of the subject (see [2], [7]) is to exploit the solvability of the operators Received by the editors November 17, 1988 and, in revised form, January 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E99; Secondary 58G99. This research was supported by NSF under MCS 87-00551AO1. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

A theorem on noncontinuable solutions is proved for abstract Volterra integral equations with operator-valued kernels (continuous and polar). It is shown that if there is no global solvability, then the C-norm of the solution is unbounded but does not tend to infinity in general. An example of Volterra equations whose noncontinuable solutions are unbounded but not infinitely large is constructed. It is shown that the theorems on noncontinuable solutions of the Cauchy problem for abstract equations of the first and nth kind (with a linear leading part) are special cases of the theorems proved in this paper.

The Sturm‐Liouville operator with singular potentials of class on a graph of arbitrary geometrical structure is considered. We study the partial inverse problem, which consists in the recovery of the potential on a boundary edge of the graph from a subspectrum under the assumption that the potentials on the other edges are known a priori. We obtain (i) the uniqueness theorem, (ii) a reconstruction algorithm, (iii) global solvability, and (iv) local solvability and stability for this inverse problem. Our method is based on reduction of the partial inverse problem on a graph to the Sturm‐Liouville problem on a finite interval with entire analytic functions in the boundary condition.

Numerous cell functions are accompanied by phenotypic changes in viscoelastic properties, and measuring them can help elucidate higher level cellular functions in health and disease. We present a high-throughput, simple and low-cost microfluidic method for quantitatively measuring the elastic (storage) and viscous (loss) modulus of individual cells. Cells are suspended in a high-viscosity fluid and are pumped with high pressure through a 5.8 cm long and 200 µm wide microfluidic channel. The fluid shear stress induces large, ear ellipsoidal cell deformations. In addition, the flow profile in the channel causes the cells to rotate in a tank-treading manner. From the cell deformation and tank treading frequency, we extract the frequency-dependent viscoelastic cell properties based on a theoretical framework developed by R. Roscoe [1] that describes the deformation of a viscoelastic sphere in a viscous fluid under steady laminar flow. We confirm the accuracy of the method using atomic force microscopy-calibrated polyacrylamide beads and cells. Our measurements demonstrate that suspended cells exhibit power-law, soft glassy rheological behavior that is cell-cycle-dependent and mediated by the physical interplay between the actin filament and intermediate filament networks.

The failure of load-bearing structures by fracture is generally important in all phases of our society. It may concern small household items as well as expensive structures of civil or space applications and accordingly may cause varying degrees of economic distress. While the state of failure is usually easily determined as either failed or completely failed, the estimation of how close to either state a structure is, poses a much more difficult problem. It is important to recognize, however, that from an engineering point of view, the latter problem is the important one because it would allow, in principle, the prediction of the conditions leading to fracture and thus to a close estimate of the service life of a structure. Inasmuch as failures by fracture involve the growth of cracks it appears that keeping track of the size of a crack in a particular structure provides a means of assessing *quantitatively* the strength prior to complete failure. If one agrees that the description of structural strength is rationalized in terms of the size of the defects, it foll0ws that one must attempt to understand the laws that govern the growth of such defects in order to predict complete failure. Fracture of materials is a complicated process which encompasses atomistic aspects, as well as microscopic and large-scale continuum mechanical considerations. Although one of these aspects should not be considered without the other we shall be concerned with the continuum-mechanical formulation of the problem of fracture growth in viscoelastic materials. From this viewpoint the prediction of failure comprises three phases: first an examination of the physical situation presented by a static or growing defect in a material, second the translation of this physically observable situation into a mathematical model which is amenable to analysis by currently available or extendable tools of mathematics, third the theoretical exploitation of the mathematical model in an attempt to predict the behavior of defects under load and the comparison of these results with experimentally observable phenomena to assess the validity of the modelling process as given from phase one and phase two. While there are many important details that have bearing on such a development we shall be concerned more with the principles of the analysis and show how the various considerations of the three phases enter into the overall structure of the crack propagation problem. In keeping analytic work as simple as possible it is intended to emphasize what type of results may be obtained with the aid of continuum mechanics and where continuum mechanics requires support by microscopic considerations.

The initial boundary-value problem for the nonlinear nonlocal Sobolev equation is studied. Sufficient conditions for local and for global (with respect to time) solvability are obtained. For the case of local (not global) solvability, upper and lower bounds for the lifespan of the solution are obtained in the form of explicit and implicit formulas.

There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.