Natural Energy Space Research Articles

Global existence and blowup of solutions to a class of wave equations with Hartree type nonlinearity

Abstract In this paper, we consider a class of wave-Hartree equations on a bounded smooth convex domain with Dirichlet boundary condition. We prove the local existence of solutions in the natural energy space by using the standard Galërkin method. The results on global existence and nonexistence of solutions are obtained mainly by means of the potential well theory and concavity method.

Synchronization phenomena in thermoelastic waves connected in parallel

In this paper, the phenomenon of synchronization in nonlinearly coupled thermoelastic waves is investigated. The existence of a finite-dimensional smooth global attractor A α is proven in the natural energy space, where α is the coupling parameter. It is also proved that the coupled thermoelastic wave system can be reduced to a single thermoelastic wave when α tends to infinity. Subsequently, the upper-semicontinuity of their attractors is proven as α → ∞ . These results are ultimately employed to establish the synchronization phenomena for coupled thermoelastic waves. To the best of our knowledge, the synchronization of thermoelastic waves has not been considered before.

Quasi-classical ground states. I. Linearly coupled Pauli–Fierz Hamiltonians

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy \mathcal{E}(u,f) of product states of the form u\otimes\Psi_f , where u is a normalized state for the particle and \Psi_f is a coherent state in Fock space for the field, gives the energy of a Klein–Gordon–Schrödinger system. We minimize the functional \mathcal{E}(u,f) on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein–Gordon–Schrödinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.

Uniform tail-ends estimates of the Navier-Stokes equations on unbounded channel-like domains

This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition.

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

Estimation of non-uniqueness and short-time asymptotic expansions for Navier–Stokes flows

There is considerable evidence that solutions to the three-dimensional Navier–Stokes equations in the natural energy space are not unique. Assuming this is the case, it becomes important to quantify how non-uniqueness evolves. In this paper we provide an algebraic estimate for how rapidly two possibly non-unique solutions can separate over a compact spatial region in which the initial data has sub-critical regularity. Outside of this compact region, the data is only assumed to be in the scaling critical weak Lebesgue space and can be large. To establish this upper bound on the separation rate, we develop a new spatially local, short-time asymptotic expansion which is of independent interest.

The slightly compressible Brinkman-Forcheimer equations and its incompressible approximation

This paper investigates the slightly compressible Brinkman–Forchheimer equations (BFEs): ∂tu−Δxu+∇xp+f(u)=g with D−1(t)∂tp+divu=0 in a bounded 3D domain with Dirichlet boundary conditions. The features of this problem is that, formally, this system is partially dissipative, and will recover to incompressible BFEs when the time-dependent coefficient D(t) goes to infinity as t→∞. The well-posedness and dissipation are obtained in the natural energy space. In addition, our result reveals that the velocity field u can be approximated by the solution uin of incompressible BFEs in L2 provided that ∂tpin is uniform bounded in L2, here pin is the pressure of incompressible BFEs.

Global attractors and synchronization of coupled critical Lamé systems with nonlinear damping

We investigate the global attractors and synchronization phenomenon of a coupled critical Lamé system defined on a smooth bounded domain Ω⊂R3 with nonlinear damping and nonlinear forces of critical growth. The existence of a unique and finitely dimensional global attractor Aϰ is proved in the natural energy space H=[D((−Δe)12)]2×[(L2(Ω))3]2, where ϰ is the coupling parameter and Δe is the Lamé operator. This attractor is further proved to be smooth in the regular space [(H2(Ω))3∩(H01(Ω))3]2×[(H01(Ω))3]2. We also show that the coupled Lamé system can be reduced to a single one when ϰ tends to infinity. Then we can compare dynamics of the coupled and single systems by proving the upper-semicontinuity of their attractors in Hδ=[(H2−δ(Ω))3∩(H01−δ(Ω))3]2×[(H01−δ(Ω))3]2 for any δ∈(0,1) as ϰ→∞. These results are finally used to study the asymptotic and exponential synchronization phenomena for the coupled Lamé system.

Asymptotic behavior of plate equations driven by colored noise on unbounded domains

This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on mathbb{R}^{n}. First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

On the global existence and blow-up for the double dispersion equation with exponential term

<abstract><p>This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively.</p></abstract>

The damped wave equation with acoustic boundary conditions and non-locally reacting surfaces

The aim of the paper is to study the problem utt+dut-c2Δu=0inR×Ω,μvtt-divΓ(σ∇Γv)+δvt+κv+ρut=0onR×Γ1,vt=∂νuonR×Γ1,∂νu=0onR×Γ0,u(0,x)=u0(x),ut(0,x)=u1(x)inΩ,v(0,x)=v0(x),vt(0,x)=v1(x)onΓ1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_{tt}+du_t-c^2\\Delta u=0 \\qquad &{}\ ext {in}\\, {\\mathbb {R}}\ imes \\Omega ,\\\\ \\mu v_{tt}- \ extrm{div}_\\Gamma (\\sigma \ abla _\\Gamma v)+\\delta v_t+\\kappa v+\\rho u_t =0\\qquad &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _1,\\\\ v_t =\\partial _\ u u\\qquad &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _1,\\\\ \\partial _\ u u=0 &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _0,\\\\ u(0,x)=u_0(x),\\quad u_t(0,x)=u_1(x) &{} \ ext {in}\\, \\Omega ,\\\\ v(0,x)=v_0(x),\\quad v_t(0,x)=v_1(x) &{} \ ext {on}\\, \\Gamma _1, \\end{array}\\right. } \\end{aligned}$$\\end{document}where Omega is a open domain of {mathbb {R}}^N with uniformly C^r boundary (Nge 2, rge 1), Gamma =partial Omega , (Gamma _0,Gamma _1) is a relatively open partition of Gamma with Gamma _0 (but not Gamma _1) possibly empty. Here textrm{div}_Gamma and nabla _Gamma denote the Riemannian divergence and gradient operators on Gamma , nu is the outward normal to Omega , the coefficients mu ,sigma ,delta , kappa , rho are suitably regular functions on Gamma _1 with rho ,sigma and mu uniformly positive, d is a suitably regular function in Omega and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when Omega is bounded, Gamma _1 is connected, r=2, rho is constant and kappa ,delta ,dge 0.

Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.

Global attractors of the degenerate fractional Kirchhoff wave equation with structural damping or strong damping

Abstract This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite by using Z 2 {{\mathbb{Z}}}_{2} index theory.

Stability of Peaked Solitary Waves for a Class of Cubic Quasilinear Shallow-Water Equations

AbstractThis paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $H^1\cap W^{1,4}$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.

Orbital Stability of Internal Waves

This paper studies the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. A diverse collection of small-amplitude solitary wave solutions for this system have been constructed by several authors in the case of strong surface tension (as measured by the Bond number) and slightly subcritical Froude number. We prove that all of these waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we prove that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE model equation. These results are obtained by reformulating the problem as an infinite-dimensional Hamiltonian system, then applying a version of the Grillakis–Shatah–Strauss method recently introduced in Varholm et al. (Commun Pure Appl Math 73:2634–2684, 2020). A key part of the analysis consists of computing the spectrum of the linearized augmented Hamiltonian at a shear flow or small-amplitude wave. For this, we generalize an idea used by Mielke (R Soc Lond Philos Trans Ser A Math Phys Eng Sci 360:2337–2358, 2002) to treat capillary-gravity water waves beneath vacuum.

Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

<abstract><p>The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.</p></abstract>

Global attractor for a wave model with nonlocal nonlinear damping

This paper investigates the well-posedness and long-time dynamics of a wave model with nonlocal nonlinear damping: utt−Δu+σ(‖∇u‖2)g(ut)+f(u)=h(x). For a new exponent p⁎=6γγ+1(≥3), where γ∈[1,5) is the growth index of the nonlinear damping term g(ut), it shows that, as the growth exponent p of the nonlinearity f(u) satisfies 2≤p≤p⁎, the problem is well-posed and has a global attractor in the natural energy space H=H01(Ω)×L2(Ω).

Well-posedness and asymptotic stability of solutions to the semilinear wave equation with analytic nonlinearity and time varying delay

We consider the semilinear wave equation with time varying delay and linear internal feedbackutt(x,t)−Δu(x,t)+μ1ut(x,t)+μ2ut(x,t−τ(t))+f(x,u)=g, in a regular bounded domain of Rn with Dirichlet boundary conditions. Under suitable relations between μ1 and μ2 and suitable conditions on f and τ we derive first a boundedness result and we show that any global bounded solution of the above equation has a relative compact range in the natural energy space. More importantly, when the nonlinear term f is analytic in the second variable, we show the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution. We also discuss existence of global solutions.

Upper semicontinuity of strong attractors for the Kirchhoff wave model with structural nonlinear damping

In this paper, we prove the upper semicontinuity of the strong global attractors on the dissipative index θ in the topology of the stronger space for the Kirchhoff wave model with structural nonlinear damping: , with θ ∈ [1/2, 1). It is continuation of the research in recent literatures1,2 where the upper semicontinuity of the strong attractor on θ in the topology of natural energy space is obtained. This result improves and deepens those in recent literatures (Li and Yang in J Differ Equat. 2020; 268: 7741‐7743; Li, Yang and Ding in Appl. Math. Lett. 2020; 104:106258).

Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $

The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x) $. It shows that when the nonlinearity $ g(u) $ is of supercritical growth $ p $, with $ \frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+} $, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as $ t>0 $; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space $ \mathbb{R}^N $ rather than approximating physical space $ \mathbb{R}^N $ by a sequence of balls $ \Omega_R $ as usual, we break through the longstanding existed restriction on this topic for $ p: 1\leqslant p\leqslant p^* $.