Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted $L^1$ spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on decay properties and asymptotic profiles of solutions. We find that the regularity-loss structure is destroyed by structural damping. Finally, we give some applications of our results on thermoelastic plate equations and damped Moore-Gibson-Thompson equation.

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations: u_{tt} + ∆^2u + ∆θ = 0 \text{ and } θ_t − ∆θ − ∆u_t = 0\quad \text{in } Ω × (0, ∞), subject to Dirichlet boundary conditions: u|_Γ = D_ν u|_Γ = θ|_Γ = 0 and initial conditions (u,u_t,θ)|_{t=0} = (u_0, v_0, θ_0) . Here Ω is an exterior domain (domain with bounded complement) in ℝ^n with n = 2 or n= 3 , the boundary Γ of which is assumed to be a C^4 -hypersurface.

This article introduces new sufficient conditions ensuring the interior approximate controllability of semilinear thermoelastic plate equations subject to Dirichlet boundary conditions. The analysis is carried out by reformulating the system as an abstract evolution equation on a suitable Banach space. A key role is played by the compactness of the semigroup generated by the linear operator, which allows us to treat the nonlinear components effectively. To establish controllability, we apply Rothe’s fixed-point theorem, which provides the necessary framework for handling nonlinear perturbations. The results obtained contribute to the existing literature, since the controllability of the specific semilinear thermoelastic system considered here has not been previously investigated.

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

We consider (linear) thermo-elastic plate equations under five sets of canonical B.C., including two cases where the mechanical and the thermal variables are coupled on the boundary. The challenging so-called free B.C. case of [Lag], [A-L] is included. The main results are as follows. If rotational forces are not accounted for, then the resulting s.c. contraction semigroup is, moreover, analytic on the natural (energy) space under all such canonical B.C. By contrast, if rotational forces are accounted for, then the corresponding s.c. contraction semigroup has a structural property that makes it more akin to a s.c. group (at least in the mechanical part); a fortiori, it is neither compact, nor differentiable, nor uniformly continuous for all t > 0. Analyticity of the s.c. thermo-elastic semigroup, particularly in the difficult case of free B.C., has been an open problem for some time in specialized circles. Similarly, a general description of the cases where analyticity fails has been the object of inquiries.

Structural stability for a linear system of thermoelastic plate equations

We are interested in studying models of thermoelasticity from a unified point of view, i.e. we are studying Cauchy problems for linear second-order systems of the form Ut + A0U + A1Ux − A2Uxx = 0 for d-dimensional unknowns U = U(t, x) with (t, x) ∈ R>0 × R and Ai being real and constant d× d matrices. Using a diagonalization procedure in phase space, we are able to obtain (under certain conditions) solution representations and to apply these to the derivation of well-posedness results, L–L decay estimates and diffusion phenomena.

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the Lq framework

When a system made of a large number of molecules is considered, the description of the dynamics of each individual member of the system is practically impossible, and it is necessary to resort to the methods of Statistical Mechanics. The chapter introduces the concept of distribution function in the phase space and provides the definition of statistical average (over the phase space and momentum space) of a dynamic variable. The derivation of the equilibrium distribution in the classical case follows, leading to the Maxwell-Boltzmann distribution. The analysis proceeds with the derivation of the continuity equation in the phase space: the collisionless case is treated first, followed by the more general case where the collisions are present, thus leading to the Boltzmann Transport Equation. In the Complements, after a discussion about the condition of a vanishing total momentum and angular momentum in the equilibrium case, and the derivation of statistical averages based on the Maxwell-Boltzmann distribution, the Boltzmann H-theorem is demonstrated. This is followed by an illustration of the apparent paradoxes brought about by Boltzmann’s Transport Equation and H-theorem: the violation of the symmetry of the laws of mechanics with respect to time reversal, and the violation of Poincare’s time recurrence. The illustration is carried out basing on Kac’s ring model. The chapter is completed by the derivation of the equilibrium limit of the Boltzmann Transport Equation.

In this paper, we consider the following perturbed nonlocal elliptic equation$\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right.$where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.

We establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line. The class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience. These systems possess a natural spatial translation symmetry, but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available, due to the infinite range forward- and backward-coupling through nonlocal convolution operators. We perform a reduction relying entirely on functional analytic methods. Despite the nonlocal nature of the problem, we do recover a local differential equation describing the dynamics on the set of small bounded solutions, exploiting that the translation invariance of the original problem induces a flow action on the center manifold. We apply our reduction procedure to problems in mathematical neuroscience, illustrating in particular the new type of algebra necessary for the computation of Taylor jets of reduced vector fields.

In this paper we investigate the large time behavior of solutions to the Cauchy problem on R for a one-dimensional thermoelastic system with dissipation. When the initial data is suitably small, (S. Zheng, Chin. Ann. Math. 8B (1987), 142-155)established the global existence and the decay properties of the solution. Our aim is to improve the results and to obtain the sharper decay properties, which seems to be optimal. The proof is given by the energy method and the Green function method.

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

In this paper, we show that the Kirchhoff equations are derived from the Schrodinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of n point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions.