Free Initial Data Research Articles

Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman–Forchheimer equations with the integral overdetermination

In this article, we study an inverse problem for the following convective Brinkman–Forchheimer (CBF) equations: $$\begin{aligned} \varvec{u}_t-\mu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\alpha \varvec{u} +\beta |\varvec{u}|^{r-1}\varvec{u}+\nabla p=\varvec{F}:=f \varvec{g}, \ \ \ \nabla \cdot \varvec{u}=0, \end{aligned}$$ in a bounded domain $$\Omega \subset \mathbb {R}^2$$ with smooth boundary $$\partial \Omega$$ , where $$\alpha ,\beta ,\mu >0$$ and $$r\in [1,3]$$ . The investigated inverse problem consists of reconstructing the vector-valued velocity function $$\varvec{u}$$ , the pressure field p and the scalar function f. For the divergence free initial data $$\varvec{u}_0 \in \mathbb {L}^2(\Omega )$$ , we prove the existence of a solution to the inverse problem for two-dimensional CBF equations with the integral overdetermination condition, by showing the existence of a unique fixed point for an equivalent operator equation (using an extension of the contraction mapping theorem). Moreover, we establish the uniqueness and Lipschitz stability results of the solution to the inverse problem for 2D CBF equations with $$r \in [1,3]$$ .

The Poisson brackets of free null initial data for vacuum general relativity

A hypersurface composed of two null sheets, or ‘light fronts’, swept out by the two congruences of future null normal geodesics emerging from a spacelike 2-disk can serve as a Cauchy surface for a region of spacetime. Already in the 1960s free (unconstrained) initial data for vacuum general relativity were found for hypersurfaces of this type. Here the Poisson brackets of such free initial data are calculated from the Hilbert action. The brackets obtained can form the starting point for a constraint free canonical quantization of general relativity and may be relevant to holographic entropy bounds for vacuum gravity. Several of the results of the present work have been presented in abbreviated form in the letter (Reisenberger 2008 Phys. Rev. Lett. 101 211101).

Discretely Self-Similar Solutions to the Navier–Stokes Equations with Besov Space Data

We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space $\dot B^{-1+3/p}_{p,\infty}$ where $3< p< 6$. We also construct discretely self-similar solutions for divergence free initial data in $\dot B^{-1+3/p}_{p,\infty}$ for $3<p<6$ that is discretely self-similar for some scaling factor $\lambda>1$. These results extend those of \cite{BT1} which dealt with initial data in $L^3_w$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$ for $p>3$. We also provide several concrete examples of vector fields in the relevant function spaces.

Integrable structures and the quantization of free null initial data for gravity

Variables for constraint free null canonical vacuum general relativity are presented which have simple Poisson brackets that facilitate quantization. Free initial data for vacuum general relativity on a pair of intersecting null hypersurfaces has been known since the 1960s. These consist of the ‘main’ data which are set on the bulk of the two null hypersurfaces, and additional ‘surface’ data set only on their intersection 2-surface. More recently the complete set of Poisson brackets of such data has been obtained. However the complexity of these brackets is an obstacle to their quantization. Part of this difficulty may be overcome using methods from the treatment of cylindrically symmetric gravity. Specializing from general to cylindrically symmetric solutions changes the Poisson algebra of the null initial data surprisingly little, but cylindrically symmetric vacuum general relativity is an integrable system, making powerful tools available. Here a transformation is constructed at the cylindrically symmetric level which maps the main initial data to new data forming a Poisson algebra for which an exact deformation quantization is known. (Although an auxiliary condition on the data has been quantized only in the asymptotically flat case, and a suitable representation of the algebra of quantum data by operators on a Hilbert space has not yet been found.) The definition of the new main data generalizes naturally to arbitrary, symmetryless gravitational fields, with the Poisson brackets retaining their simplicity. The corresponding generalization of the quantization is however ambiguous and requires further analysis.

Asymptotic structure of the null surface formulation and the classical graviton

The dynamical equations for the null surface formulation of general relativity for purely radiative spacetimes are derived. Those asymptotically flat spacetimes describe the nonlinear evolution of gravitational radiation and represent a classical graviton. The evolution equations constitute a set of three partial differential equations in a six-dimensional space and the source term is the free initial data of incoming gravitational radiation. The Huygens part of the wave propagation, backreaction terms, and source terms are identified in the resulting equations. An analysis of the range of validity of these equations based on the development of caustics is also given.

Characteristic Initial Data and Smoothness of Scri. I. Framework and Results

We analyze the Cauchy problem for the vacuum Einstein equations with data on a complete light-cone in an asymptotically Minkowskian space-time. We provide conditions on the free initial data which guarantee existence of global solutions of the characteristic constraint equations. We present necessary-and-sufficient conditions on characteristic initial data in $3+1$ dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.

Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data

Liquid crystalline polymers have been extensively studied in shear starting from an equilibrium nematic phase. In this study, we explore the transient and long-time behavior as a steady shear cell experiment commences during an isotropic-nematic (I-N) phase transition. We initialize a localized Gaussian nematic droplet within an unstable isotropic phase with nematic, vorticity-aligned equilibrium at the walls. In the absence of flow, the simulation converges to a homogeneous nematic phase, but not before passing through quite intricate defect arrays and patterns due to physical anchoring, the dimensions of the shear cell, and transient backflow generated around the defect arrays during the I-N transition. Snapshots of this numerical experiment are then used as initial data for shear cell experiments at controlled shear rates. For homogeneous stable nematic equilibrium initial data, the Leal group [4, 5, 6] and the authors [12] confirm the Larson-Mead experimental observations [7, 8]: stationary 2-D roll cells and defect-free 2-D orientational structure at low shear rates, followed at higher shear rates by an unstable transition to an unsteady 2-D cellular flow and defect-laden attractor. We show at low shear rates that the memory of defect-laden data lasts forever; 2-D steady attractors of [4, 5, 12] emerge for defect free initial data, whereas 1-D unsteady attractors arise for defect-laden initial data.

The Poisson Bracket on Free Null Initial Data for Gravity

Free initial data for general relativity on a pair of intersecting null hypersurfaces are well known, but the lack of Poisson brackets and concerns about caustics have stymied the development of a constraint free canonical theory. Here it is pointed out how caustics and generator crossings can be neatly avoided and Poisson brackets on free data are given. On sufficiently regular functions of the solution spacetime geometry these brackets match the Poisson brackets defined on such functions by the Hilbert action via Peierls' prescription. The symplectic 2-form is also given in terms of free data.

Gauge-invariant and coordinate-independent perturbations of stellar collapse: The interior

Small non-spherical perturbations of a spherically symmetric but time-dependent background spacetime can be used to model situations of astrophysical interest, for example the production of gravitational waves in a supernova explosion. We allow for perfect fluid matter with an arbitrary equation of state p=p(rho,s), coupled to general relativity. Applying a general framework proposed by Gerlach and Sengupta, we obtain covariant field equations, in a 2+2 reduction of the spacetime, for the background and a complete set of gauge-invariant perturbations, and then scalarize them using the natural frame provided by the fluid. Building on previous work by Seidel, we identify a set of true perturbation degrees of freedom admitting free initial data for the axial and for the l>1 polar perturbations. The true degrees of freedom are evolved among themselves by a set of coupled wave and transport equations, while the remaining degrees of freedom can be obtained by quadratures. The polar l=0,1 perturbations are discussed in the same framework. They require gauge fixing and do not admit an unconstrained evolution scheme.

Gravitational fields near space-like and null infinity

Near space-like infinity an initial value problem for the conformal Einstein equations is formulated such that: (i) the data and equations are regular, (ii) space-like and null infinity have a finite representation, with their structure and location known a priori, and (iii) the setting relies entirely on general properties of conformal structures. A first analysis of this problem shows that the solutions develop in general a certain type of logarithmic singularity at the set where null infinity touches space-like infinity. These singularities form an intrinsic part of the solutions' conformal structure. Conditions on the free initial data near space-like infinity are derived which ensure that for solutions developing from these data singularities of this type cannot occur.

Dual-null dynamics of the Einstein field

The Einstein gravitational field is decomposed with respect to two intersecting foliations of null surfaces. The dynamics are cast in the form of suitably generalized Lagrangian and Hamiltonian theories. The vacuum Einstein equations are written in terms of geometrical fields and operators adapted to the null foliations. There are no constraints. The free initial data are identified.

The general solution to the Einstein equations on a null surface

On a null 3-surface, the general solution to the vacuum Einstein equations is found in closed form in terms of free initial data. The solution is valid up to caustics, whose occurrence is determined by a necessary and sufficient condition on the initial data.

On purely radiative space-times

The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein's vacuum field equations possessing a smooth structure at past null infinity which forms the “future null cone at past timelike infinity with complete generators.” The “pure radiation problem” is analysed where “free initial data” for Einstein's field equations are prescribed on the null cone at past time-like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.