Spatial Decay Estimates for a Coupled System of Second-Order Quasilinear Partial Differential Equations Arising in Thermoelastic Finite Anti-plane Shear

The spatial decay behavior of solutions of a coupled system of second-order quasilinear partial differential equations, in divergence form, defined on a two-dimensional semi-infinite strip, is investigated. Such equations arise in the theory of anti-plane shear deformations for isotropic nonlinearly thermoelastic solids. Differential inequality techniques are employed to obtain exponential decay estimates. The results are illustrated by several examples. The results are relevant to Saint-Venant principles for nonlinear thermoelasticity as well as to theorems of Phragmen-Lindelof type. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some qualitative results for a modification of the Green–Lindsay thermoelasticity

In this short note we consider a recent modification of the Green–Lindsay thermoelastic theory proposed at Yu et al. (Meccanica 53:2543–2554, 2018). We consider a functional defined on the solutions of the problem. It allows us to obtain the continuous dependence of the solutions with respect to the initial conditions and to the supply terms, the time exponential decay of solutions and an alternative of Phragmén–Lindelöf type for the spatial behaviour.     

Coupled radially symmetric linear thermoelasticity

In this study we consider linear thermoelastic wave propagation with second sound. We consider two theories; a theory based on the Maxwell-Cataneo relation and a linearized theory based on a simplified form of a generalization of classical thermoelasticity. We consider cylindrically and spherically symmetric longitudinal waves, and for both problems we obtain expressions for the initial discontinuities, and also the time rate of decay of propagating discontinuities. Numerical solutions are obtained from the application of the method of characteristics, and further, a technique is proposed which allows numerical solutions, valid for times large compared with the relaxation time, to be efficiently generated.     

Theory of thin thermoelastic rods made of porous materials

In this paper, we consider thin rods modeled by the direct approach, in which the rod-like body is regarded as a one-dimensional continuum (i.e., a deformable curve) with a triad of rigidly rotating orthonormal vectors attached to each material point. In this context, we present a model for porous thermoelastic curved rods, having natural twisting and arbitrary shape of cross-section. To describe the porosity, we employ the theory of elastic materials with voids. The basic laws of thermodynamics are applied directly to the one-dimensional continuum, and the nonlinear governing equations are established. We formulate the constitutive equations and determine the structure of constitutive tensors. We prove the uniqueness of solution to the boundary-initial-value problem associated with the deformation of porous thermoelastic rods in the framework of linear theory. Then, we show the decoupling of the bending-shear and extension-torsion problems for straight porous rods. Using a comparison with three-dimensional equations, we identify and give interpretations to the relevant fields introduced in the direct approach. Finally, we consider the case of orthotropic materials and determine the constitutive coefficients for deformable curves in terms of three-dimensional constitutive constants by means of comparison between simple solutions obtained in the two approaches for porous thermoelastic rods.     

Rich dynamics in a non-local population model over three patches

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of non-negative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, including bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

On Singular Limits of Mean-Field Equations

Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.     

Existence,Uniqueness, and Asymptotic Stability for a Thermoelastic Plate

In this note, we are concerned with the linear theory of a thermoelastic plate when a rate-type equation is assumed for the heat flux. We consider an initial boundary-value problem for this plate and show the existence, uniqueness, and asymptotic stability of a solution. Thermodynamic restrictions on the assumed constitutive equations are also derived. Finally, we give an expression for the pseudo-free energy.     

Universal equations of three-dimensional boundary layer theory

We consider a parametric method for investigating three-dimensional laminar motion of an incompressible fluid in a boundary layer on a curved surface. It is found that the problem solution in the general case depends on four series of parameters, constructed from two components of the outer flow velocity and the two Lamé coefficients characterizing the shape of the immersed surface. From the general equations of the three-dimensional boundary layer we obtain a system of two universal equations which do not contain the characteristics of the outer flow. This system may be solved once and for all. As an example we consider the problem of the laminar boundary layer on the walls of an axisymmetric channel in the case of swirling outer flow. For this case we obtain numerical solutions of the system of universal equations in the local two-parameter approximation.     

A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.     

On the Investigation of One Semiexplicit System of Differential Equations in the Case of a Variable Pencil of Matrices

We investigate a semiexplicit Cauchy problem for a system of ordinary differential equations in the case of a variable pencil of matrices. We determine sufficient conditions of existence of solutions and consider the problem of their number.     

Well-posedness of the equations of generalized thermoelasticity

In this paper the local existence, uniqueness and continuous dependence for smooth solutions to the initial value problem for a class of generalized (dependent on the time derivative of temperature) thermoelastic materials is proved. The field equations are written as a quasilinear hyperbolic system and the known results by Hughes, Kato and Marsden are applied.     

Incompressible Euler Equations and the Effect of Changes at a Distance

Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.     

The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations

We consider linear nonautonomous second order parabolic equations on bounded domains subject to Dirichlet boundary condition. Under mild regularity assumptions on the coefficients and the domain, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. Our main theorem extends in a natural way standard results on principal eigenvalues and eigenfunctions of elliptic and time-periodic parabolic equations. Similar theorems were earlier available only for smooth domains and coefficients. As a corollary of our main result, we obtain the uniqueness of positive entire solutions of the equations in     

On steady distributions of self-attracting clusters under friction and fluctuations

We consider the Vlasov-Fokker-Planek equation with a Newtonian, attracting potential and study its stationary solutions, given by the generalized Lane-Emden equation. In a two-dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse. For a one-dimensional model we prove some results on existence, uniqueness, stability and symmetry-breaking of stationary solutions.     

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. 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. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

Large Time Asymptotics for Partially Dissipative Hyperbolic Systems

This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover.     

Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

We consider weak solutions to a simplified Ericksen–Leslie system of two-dimensional compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity \({|\nabla\mathbf{d}|^2\mathbf{d}}\) in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.     

Global existence and exponential stability of solutions to thermoelastic equations of hyperbolic type

In this paper we prove the global existence and exponential stability of solutions to thermoelastic equations of hyperbolic type provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially. Moreover, the global solution, together with its the third-order full energy, is exponentially stable for any t > 0.     

Decay rates of higher-order norms of solutions to the Navier-Stokes-Landau-Lifshitz system

In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the global solutions with the help of an energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solutions by applying a Fourier splitting method introduced by Schonbek(SCHONBEK, M. E. L~2 decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 88, 209–222(1985)) under an additional assumption that the initial perturbation is bounded in L~1(R~3).