Natural Energy Space Research Articles

Very weak solutions of degenerate parabolic systems with non-standard [formula omitted]-growth

We study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p(x,t)-Laplacian system, ∂tu−div(|Du|p(x,t)−2Du)=div(|F|p(x,t)−2F). Under natural assumptions on the exponent function p:Ω×(0,T)→[2,∞), we prove that any very weak solution u:Ω×(0,T)→RN with |Du|p(⋅)(1−ε)∈L1 belongs to the natural energy space, i.e. |Du|p(⋅)∈Lloc1, provided ε>0 is small enough.

Well-posedness and asymptotic behavior of a nonautonomous, semilinear hyperbolic-parabolic equation with dynamical boundary condition of memory type

We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term $f$ is real analytic, we then derive an appropriate Lyapunov energy and we use the {\L}ojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.

An Lp theory for stationary radiative transfer

We present a self-contained analysis of the stationary radiative transfer equation in weighted spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for under minimal assumptions on the parameters. By constructing an explicit example, we show that our estimates are sharp and cannot be improved in general. Better estimates are, however, derived under additional assumptions on the parameters. We also present estimates for derivatives and traces of the solution and formulate a natural energy space, for which the data-to-solution map becomes an isomorphism. As a side result, we are able to prove uniform convergence of the source iteration for all without the assumption of positive absorption that is frequently used in the literature.

Existence and Asymptotic Behavior of Solutions to Semilinear Wave Equations with Nonlinear Damping and Dynamical Boundary Condition

Our aim in this article is to study a nonautonomous semilinear wave equation with nonlinear damping and dynamical boundary condition. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Then, by deriving an appropriate Lyapunov energy, we show that if the exponent in the Łojasiewicz-Simon inequality is large enough (depending on the damping), then weak solutions converge to equilibrium.

Limiting behavior of blow-up solutions of the NLSE with a stark potential

This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear Schrödinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the limiting behavior of blow-up solutions with critical and small super-critical mass are obtained in the natural energy space ∑={u∈H1;∫ℝN|x|2|u|2dx<+∞}.. Moreover, an interesting concentration property of the blow-up solutions with critical mass is gotten, which reads that |u|(t,x)|2→||Q||L22δx=x1 as t →T.

Long-term analysis of strongly damped nonlinear wave equations

We consider the strongly damped nonlinear wave equation with Dirichlet boundary conditions, which serves as a model in the description of thermal evolution within the theory of type III heat conduction. In particular, the nonlinearity f acting on ut is allowed to be nonmonotone and to exhibit a critical growth of polynomial order 5. The main focus is the long-term analysis of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space.

AN EVOLUTIONARY ELASTOPLASTIC PLATE MODEL DERIVED VIA Γ-CONVERGENCE

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff–Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Γ-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies.

Long-time dynamics of Kirchhoff wave models with strong nonlinear damping

We study well-posedness and long-time dynamics of a class of quasilinear wave equations with a strong damping. We accept the Kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the L 2 -norm of the gradient of the displacement. We prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration (or even negativity) of the stiffness coefficient and the case of a supercritical source term. Our main results deal with global attractors. For strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite-dimensional attractor. In the non-supercritical case this attractor is strong. In this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals.

Compact decoupling for an abstract system of thermoelasticity of type III

In this note, we show that the difference between the C 0 -semigroups generated by an abstract system of thermoelasticity of type III and a suitable decoupled system is compact in the underlying natural energy space. The problem is reduced to prove the norm continuity of this difference and the compactness of the difference between the resolvent of their generators. The argument is based on an operator semigroup technique.

Backwards uniqueness of the $C_{0}$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system

In this paper the “backward-uniqueness property” is ascertained for a two- or three-dimensional, fluid-structure interactive partial differential equation (PDE) system, for which an explicit $C_{0}$-semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space $\mathbf {H}$. (See also their Contemporary Mathematics article of 2007 for a preliminary, simplified, canonical model.) This system of coupled PDEs comprises the parabolic Stokes equations and the hyperbolic Lamé system of dynamic elasticity. Each dynamic evolves within its respective domain, while being coupled on the boundary interface between fluid and structure. In terms of said fluid-structure semigroup $\left \{ e^{\mathcal {A}t}\right \}$, posed on the associated finite energy space $\mathbf {H}$, the backward-uniqueness property can be stated in this way: If for given initial data $y_{0}\in \mathbf {H}$, $e^{\mathcal {A} T}y_{0}=0$ for some $T>0$, then necessarily $y_{0}=0$. The proof of this property hinges on establishing necessary PDE estimates for a certain static fluid-structure equation in order to invoke the abstract backward-uniqueness resolvent-based criterion by Lasiecka, Renardy, and Triggiani (2001). The backward-uniqueness property for the coupled Stokes-Lamé PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables $\{w,w_t\}$) and, simultaneously, approximate controllability (in the parabolic state variable $u$) of the present coupled PDE model, under boundary control. A similar situation occurred for thermoelastic models as shown in papers by M. Eller, V. Isakov, I. Lasiecka, M. Renardy, and R. Triggiani.

Energy-Critical NLS with Quadratic Potentials

We consider the defocusing -critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential . We show global well-posedness for radial initial data obeying ∇u 0(x), xu 0(x) ∈ L 2. In view of the potential V, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.

Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

This article is focused on an established, genuinely physical fluid-structure interaction model, whereby the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic–hyperbolic system of two partial differential equations in three dimensions with non-standard coupling at the boundary interface: the (dynamic) Stokes system (parabolic, modelling the fluid) and the Lamé system (hyperbolic, modelling the structure). This system generates a contraction semigroup on the natural energy space [G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties, Fluids and Waves, Amer. Math. Soc. Contemp. Math. 440 (2007), pp. 15–59] (canonical model) and [G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Series S, 2(3) (2009), pp. 417–447]. The boundary interface may or may not include a ‘damping’ (or dissipative) term. If damping is active on the entire interface, then uniform (exponential) stabilization is ensured, regardless of the geometry of the structure [G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22(4) 2008, pp. 817–835, special issue, invited paper] (canonical model) and [G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system, J. Evol. Eqns 9(2009), pp. 341–370]. This article emphasizes the case of, at most, partial damping. At any rate, the main result is a precise uniform-operator limit behaviour of the resolvent operator of the semigroup generator on the imaginary axis of interest in itself, which holds true with or without damping. It, in turn, then implies a fortiori strong stability results: most notably, on the whole state space, under at least partial damping at the interface; and, in the absence of damping, on the whole state space, after factoring out an explicit one-dimensional null eigenspace, at least for a large class of geometries of the structure: these are characterized by a uniqueness property of a special over-determined elliptic problem.

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.

Offshore wind power development as affected by seascape values on the German North Sea coast

In Germany, the first permits have now been issued for the construction of large-scale offshore wind farms in the German Exclusive Economic Zone (EEZ). This paper focuses on perceptions of the local seascape and the role of aesthetic seascape qualities in shaping local attitudes to offshore wind farming. Based on a survey of local residents in the districts of Dithmarschen and North Frisia, it shows that aesthetic seascape perception alone cannot account for local attitudes to offshore wind farming. Three main aspects seem to come together to determine these attitudes: deeply held convictions of the sea as a natural space, deeply held views of the local landscape and linked to this local identity, and also perceptions of renewable energies in combination with attitudes to issues such as climate change and sea level rise. The paper draws some conclusions on the future of the sea as a natural space or energy space.

Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

We consider a parabolic–hyperbolic coupled system of two partial differential equations (PDEs), which governs fluid–structure interactions, and which features a suitable boundary dissipation term at the interface between the two media. The coupled system consists of Stokes flow coupled to the Lame system of dynamic elasticity, with the respective dynamics being coupled on a boundary interface, where dissipation is introduced. Such a system is semigroup well-posed on the natural finite energy space (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). Here we prove that, moreover, such semigroup is uniformly (exponentially) stable in the corresponding operator norm, with no geometrical conditions imposed on the boundary interface. This result complements the strong stability properties of the undamped case (Avalos and Triggiani in Discr Contin Dynam Sys, to appear).

Global attractors for semilinear damped wave equations with critical exponent on [formula omitted

Long-time dynamical properties for a class of semilinear damped wave equations with critical exponent on unbounded domain R n are studied. The existence of compact global attractors for these equations in natural energy space is proved. These attractors are characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

Global existence for parabolic problems involving the $p$-Laplacian and a critical gradient term

We study existence and regularity of solutions for nonlinear parabolic problems whose model is u t - div(|∇u| P-2 ∇u) = β(u) |∇u| P + f in 0 x ]0, ∞[, (0.1) u(x,t)=0, on ∂Ω × ]0, ∞[, u(x,0) = u 0 (x), in Ω, where p > 1 and Q C R N is a bounded open set; as far as the function β is concerned, we make no assumption on its sign; instead, we consider three possibilities of growth for β, which essentially are: (1) constant, (2) polynomial and (3) exponential. In each case, we assume appropriate hypotheses on the data f and u 0 , depending on the growth of β, and prove that a solution u exists such that an exponential function of u belongs to the natural Sobolev energy space. Since the solutions may well be unbounded, one cannot use sub/supersolution methods. However we show that, under slightly stronger assumptions on the data, the solution that we find is bounded. Our existence results, in the cases (2) and (3) above, rely on new logarithmic Sobolev inequalities.

Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface

We consider a coupled parabolic-hyperbolic PDE system arising in fluid-structure interaction, with boundary dissipation at the interface, $d = 2,3$. Such a system is semigroup well-posed in the natural energy space [4]. We then establish that it is also uniformly (exponentially) stable, thus complementing the strong stability results of the undamped case [2], [4].

On finite energy solutions of the KP-I equation

We prove that the flow map of the Kadomtsev–Petviashvili-I (KP-I) equation is not uniformly continuous on bounded sets of the natural energy space.

The Cauchy problem for the Gross–Pitaevskii equation

We prove global wellposedness of the two-dimensional and three-dimensional Gross–Pitaevskii equations in the natural energy space.