Local Energy Estimates Research Articles

Localized energy estimates for wave equations on high-dimensional Schwarzschild space-times

The localized energy estimate for the wave equation is known to be a fairly robust measure of dispersion. Recent analogs on the $(1+3)$-dimensional Schwarzschild space-time have played a key role in a number of subsequent results, including a proof of Price's law. In this article, we explore similar localized energy estimates for wave equations on $(1+n)$-dimensional hyperspherical Schwarzschild space-times.

Partial regularity for the Navier-Stokes-Fourier system

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved for each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.

The thin film equation with backwards second order diffusion

We focus on the thin film equation with lower order “backwards” diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a “porous media cutoff” of van der Waals forces. We treat in detail the equation u_t + \{u^n(u_{xxx}+ν u^{m–n}u_x–Au^{M–n}u_x)\}_x = 0, where ν = \pm 1 , n > 0 , M > m , and A ≥ 0 . Global existence of weak nonnegative solutions is proven when m–n > –2 and A > 0 or ν = –1 , and when –2 < m–n < 2 , A = 0 , ν = 1 . From the weak solutions, we get strong entropy solutions under the additional constraint that m–n > –3/2 if ν = 1 . A local energy estimate is obtained when 2 ≤ n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2 , for the case of “strong slippage”, 0 < n < 2 , when ν = 1 based on local entropy estimates, and for the case of “weak slippage”, 2 ≤ n < 3 , when ν = \pm 1 based on local entropy and energy estimates.

On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.

Local energy estimates for the finite element method on sharply varying grids

Local energy error estimates for the nite element method for el- liptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local ap- proximation term, plus a global \pollution term that measures the inuence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result.

An Assessment of Bulk and Local Stored Energy Measurements in Ultrafine Grained Interstitial Free Steel

The evolution of stored energy and associated thermal behaviour was investigated for an ultrafine grained Ti-IF steel severely deformed by Equal Channel Angular Pressing (ECAP) followed by cold rolling at ambient and liquid nitrogen temperatures. Bulk stored energy measurements by Differential Scanning Calorimetry (DSC) returned 350-600 whereas local stored energy estimates from microhardness, Electron Back-Scattering Diffraction (EBSD) and X-ray line profile analysis resulted in 5-140 . Higher bulk stored energy values correspond to the enthalpy release from all sources of strain in the material volume as well as Ti precipitation during annealing while the lower local stored energy range alludes only to dislocation content or internal stresses. An apparent activation energy of 500-550 suggests sluggish recrystallisation due to excess of Ti in solid solution.

Asymptotic estimates to quenching solutions of heat equations with weighted absorptions

This paper studies heat equations with weighted nonlinear absorptions of the form ut = uxx − Mf (x)u −p in (−1, 1) × (0, T ) subject to Dirichlet boundary conditions u(−1, t) = u(1, t) = 1 and initial data φ(x). The asymptotic estimates to quenching time and set of solutions as M → +∞ is established by local energy estimates. It is obtained that the quenching time T ∼ m p+1 · M −1 with m = 1 maxx(f (x)/φp+1(x)) as M → +∞. It is shown also how the quenching set concentrates near the maximum points of f/ φ p+1 for large M.

Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles

We prove global existence for quasilinear wave equations in high dimensional exterior domains with Dirichlet boundary conditions. In particular, we permit the nonlinear term to depend on the solution, not just its first and second derivatives. The key estimates are variants on localized energy estimates.

Stochastic NonLinear Wave Equations in Local Sobolev Spaces

Existence of weak solutions of stochastic wave equations with nonlinearities of a critical growth driven by spatially homogeneous Wiener processes is established in local Sobolev spaces and local energy estimates for these solutions are proved. A new method to construct weak solutions is employed.

Strichartz Estimates on Schwarzschild Black Hole Backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

Stabilization of Schrödinger equation in exterior domains

We prove uniform local energy estimates of solutions to the damped Schrodinger equation in exterior domains under the hypothesis of the Exterior Geometric Control. These estimates are derived from the resolvent properties.

Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions

We study the Dirichlet problem for the elliptic equations − ∑ i D i ( a i ( x , u ) | D i u | p i ( x ) − 2 D i u ) + c ( x , u ) | u | σ ( x ) − 2 u = f ( x ) in a bounded domain Ω ⊂ R n , and the class of elliptic systems − ∑ j D j ( a i j ( x , ∇ u ) ) = f ( i ) ( x , u ) , i = 1 , … , n , u = ( u ( 1 ) , … , u ( n ) ) , satisfying the growth condition ∀ ( x , s , V ) ∈ Ω × R n 2 ∑ i j a i j ( x , V ) ⋅ V i j ≥ a 0 ∑ i j | V i j | p i j ( x ) , a 0 = const > 0 . The exponents p i j ( x ) , p i ( x ) , σ ( x ) are known functions. These equations are usually referred to as elliptic equations with nonstandard growth conditions. We prove first the theorems of existence of (bounded) weak solutions and establish sufficient conditions of uniqueness of a weak solution. Our main purpose is the study of the localization properties of weak solutions: we show that the weak solution may identically vanish on a set of nonzero measure in Ω (a dead core) and derive estimates on the size and location of these dead cores in terms of the problem data. The study of the localization properties is performed via the method of local energy estimates.

LOCAL ENERGY ESTIMATES FOR THE MAXWELL–LANDAU–LIFSHITZ SYSTEM AND APPLICATIONS

We study the Maxwell–Landau–Lifshitz system without exchange energy. First, we prove an [Formula: see text] estimate for the linear wave equation and apply this local energy estimate to obtain a bound on the curl of the electromagnetic field, uniformly in time and locally in space. Next, we prove strong convergence results, when the time t tends to ∞ or when the speed of light tends to ∞ (which corresponds to the quasi-stationary approximation). Finally, we establish a stability result with respect to the damping parameter of the Landau–Lifshitz equation.

Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, invariance with respect to changes in independent and dependent variables is investigated, as are results in the classification of continuous one-parameter symmetry groups. For the variational and divergence symmetries, conservation laws are derived via the method of multipliers. The conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems, and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems.

Doubly Nonlinear Thin-Film Equations in One Space Dimension

We consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative. The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the “complete wetting” regime. Relying solely on global and local energy estimates and on Bernis’ inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support. A necessary condition for the occurrence of waiting-time phenomena is also derived.

Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains

As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u tt+u t=Δu+F(x, u) in R N , where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H1 ul(R N )×L2 ul(R N ). If N≤2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions.

Lifshitz Tail for 2D Discrete Schrödinger Operator with Random Magnetic Field

Lifshitz tail for 2 dimensional discrete Schrodinger operator with Anderson-type random magnetic field is proved. We first prove local energy estimates for deterministic discrete magnetic Schrodinger operators, and then follow the large deviation argument of Simon [6].

Integrability and boundedness of local solutions to doubly degenerate quasilinear parabolic equations

Integral and supremum estimates are proven for local solutions of doubly degenerate quasilinear parabolic equations. This is done with the aid of a local energy inequality and estimates in $L^{\text{weak}}_q$ spaces.

Integrability and boundedness of local solutions to singular and degenerate quasilinear parabolic equations

Integral and supremum estimates are proven for local solutions of degenerate and singular quasilinear parabolic equations. This is done with the aid of a local energy inequality and estimates in $L^{\text{weak}}_q$ spaces.