Finite Energy Space Research Articles

An inf-sup approach to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup generation for an interactive composite structure-Stokes PDE dynamics

In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered)–fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems. By means of a nonstandard mixed variational formulation, we show that the PDE system generates a C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup on the associated finite energy space of data. In this work, the presence of the pressure term in the 3D Stokes equation adds a great challenge to our analysis. To overcome this difficulty, we follow a methodology which is based on the necessarily non-Leray-based elimination of the associated pressure term, via appropriate nonlocal operators. Moreover, while we express the fluid solution variable via decoupling of the Stokes equation, we construct the elastic solution variables by solving a mixed variational formulation via a Babuska–Brezzi approach.

The scattering map on collapsing charged spherically symmetric spacetimes

In this paper we generalise our previous results (Alford 2020 Ann. Inst. Henri Poincare 21 2031–92) concerning scattering on the exterior of collapsing dust clouds to the charged case, including in particular the extremal case. We analyse the energy boundedness of solutions φ to the wave equation on the exterior of collapsing spherically symmetric charged matter clouds. We then proceed to define the scattering map on this spacetime, and look at the implications of our boundedness results on this map. More specifically, we first construct a class of spherically symmetric charged collapsing matter cloud exteriors, and then consider solutions to the wave equation with Dirichlet (reflective) boundary conditions on the surface of these clouds. We then show that the energy of φ remains uniformly bounded going forwards or backwards in time, and that the scattering map is bounded going forwards but not backwards. Therefore, the scattering map is not surjective onto the space of finite energy on I+∪H+ . Thus there does not exist a backwards scattering map from finite energy radiation fields on I+∪H+ to finite energy radiation fields on I− for these models. These results will be used to give a treatment of Hawking radiation in a companion paper (Alford 2023 arXiv:2309.03022).

Conserved energies for the one dimensional Gross-Pitaevskii equation: Low regularity case

We construct a family of conserved energies for the one dimensional Gross-Pitaevskii equation, but in the low regularity case (in [14] we have constructed conserved energies in the high regularity situation). This can be done thanks to regularization procedures and a study of the topological structure of the finite-energy space. The asymptotic (conserved) phase change on the real line with values in R/2πZ is studied. We also construct a conserved quantity, the renormalized momentum H1 (see Theorem 1.3), on the universal covering space of the finite-energy space.

Weak and strong semigroups in structural acoustic Kirchhoff-Boussinesq interactions with boundary feedback

We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.

Wellposedness, spectral analysis and asymptotic stability of a multilayered heat-wave-wave system

In this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a “thin” layer and a “thick” layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay.Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a C0-semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the C0-semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator A and show that the spectrum of A does not intersect the imaginary axis.

The Scattering Map on Oppenheimer\u2013Snyder Space-Time

In this paper, we analyse the boundedness of solutions phi of the wave equation in the Oppenheimer–Snyder model of gravitational collapse in both the case of a reflective dust cloud and a permeating dust cloud. We then proceed to define the scattering map on this space-time and look at the implications of our boundedness results on this scattering map. Specifically, it is shown that the energy of phi remains uniformly bounded going forwards in time and going backwards in time for both the reflective and the permeating cases. It is then shown that the scattering map is bounded going forwards, but not backwards. Therefore, the scattering map is not surjective onto the space of finite energy on mathcal {I}^+cup mathcal {H}^+. Thus, there does not exist a backwards scattering map from finite energy radiation fields on mathcal {I}^+cup mathcal {H}^+ to finite energy radiation fields on mathcal {I}^-. We will then contrast this with the situation for scattering in pure Schwarzschild.

Lp Metric Geometry of Big and Nef Cohomology Classes

Let (X,ω) be a compact Kahler manifold of dimension n, and let 𝜃 be a closed smooth real (1,1)-form representing a big and nef cohomology class. We introduce a metric dp,p ≥ 1, on the finite energy space $\mathcal {E}^{p}(X,\theta )$ , making it a complete geodesic metric space.

A linearized viscous, compressible flow-plate interaction with non-dissipative coupling

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile U. In contrast to model in [6], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a C0-semigroup {eAt}t≥0 on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator A, yielding the flow-structure dynamics.

The Long-Time Behavior of Modified Calabi Flow

In this paper, we study the long-time behavior of modified Calabi flow to study the existence of generalized Kahler–Ricci soliton. We first give a new expression of the modified K-energy and prove its convexity along weak geodesics. Then we extend this functional to some finite energy spaces. After that, we study the long-time behavior of modified Calabi flow.

$L^1$ metric geometry of big cohomology classes

Suppose $(X,\omega)$ is a compact Kahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the Kahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nystrom to the big case, we show that one can construct geodesic rays in this space in a flexible manner.

Feedback stabilization of a linear hydro-elastic system

It is known that the linear Stokes-Lamé system can be stabilized by a boundary feedback in the form of a dissipative velocity matching on the common interface [5]. Here we consider feedback stabilization for a generalized linear fluid-elasticity interaction, where the matching conditions on the interface incorporate the curvature of the common boundary and thus take into account the geometry of the problem. Such a coupled system is semigroup well-posed on the natural finite energy space [13], however, the system is not dissipative to begin with, which represents a key departure from the feedback control analysis in [5]. We prove that a damped version of the general linear hydro-elasticity model is exponentially stable. First, such a result is given for boundary dissipation of the form used in [5]. This proof resolves a more complex version, compared to the classical case, of the weighted energy methods, and addresses the lack of over-determination in the associated unique continuation result. The second theorem demonstrates how assumptions can be relaxed if a viscous damping is added in the interior of the solid.

Lipschitz metric for the periodic second-order Camassa–Holm equation

In this study, we consider the Cauchy problem for the second-order Camassa–Holm equation with periodic initial data u0. Using the vanishing viscosity method, the local weak solution of the equation is obtained in the finite energy space. A continuous semigroup of weak conservative solutions in Lagrangian coordinates is constructed. In particular, for any two solutions u(t) and v(t) of the equation, a Lipschitz metric dD is constructed with the property that dD(u(t),v(t))≤eCtdD(u0,v0).

Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source:{utt−k(0)Δu−∫0∞k′(s)Δu(t−s)ds+|ut|m−1ut=|u|p−1u, in Ω×(0,T),u(x,t)=u0(x,t), in Ω×(−∞,0], where Ω is a bounded domain in R3 with a Dirichlét boundary condition. The relaxation kernel k is monotone decreasing and k(∞)=1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p>max⁡{m,k(0)}, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to −∞, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.

Approximation of the controls for the linear beam equation

This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted \(L^2\)-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.

Approximation of the controls for the beam equation with vanishing viscosity

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modeling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.

Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid

We study well-posedness and long time dynamics of a coupled model consisting of the compressible isothermal linearized Navier–Stokes fluid in some 3D domain and a nonlinear elastic plate placed on a part of the boundary. We first prove the existence and uniqueness of solutions of the corresponding initial–boundary value problem in a finite energy space. Our main result states the existence of a global finite-dimensional attractor even in the absence of a mechanical damping applied to the plate.

On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

Attractors for non-dissipative irrotational von Karman plates with boundary damping

Long-time behavior of solutions to a von Karman plate equation is considered. The system has an unrestricted first-order perturbation and a nonlinear damping acting through free boundary conditions only.This model differs from those previously considered (e.g. in the extensive treatise (Chueshov and Lasiecka, 2010 [11])) because the semi-flow may be of a non-gradient type: the unique continuation property is not known to hold, and there is no strict Lyapunov function on the natural finite-energy space. Consequently, global bounds on the energy, let alone the existence of an absorbing ball, cannot be a priori inferred. Moreover, the free boundary conditions are not recognized by weak solutions and some helpful estimates available for clamped, hinged or simply-supported plates cannot be invoked.It is shown that this non-monotone flow can converge to a global compact attractor with the help of viscous boundary damping and appropriately structured restoring forces acting only on the boundary or its collar.

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.

Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources

We study the global well-posedness of the nonlinear wave equation $$ u_{tt} - \Delta u - \Delta _p u_t = f(u) $$ in a bounded domain ${\Omega} \subset \mathbb{R}^n$ with Dirichlét boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H^1_0({\Omega})$ into $L^2({\Omega})$. The nonlinear term $- \Delta _p u_t $ is a strong damping where the $-\Delta _p$ denotes the p-Laplacian (defined below). Under suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution depends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.