Initial Cauchy Data Research Articles

Inverse coefficient problems for one-dimensional time-fractional diffusion equations

We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.

Energy behavior for Sobolev solutions to viscoelastic damped wave models with time‐dependent oscillating coefficient

AbstractIn this work, we study the asymptotic behavior of the structurally damped wave equations arising from the viscoelastic mechanics. We are particularly interested in the complicated interaction of the time‐dependent oscillating coefficients on the Dirichlet Laplacian operator and the structurally damped terms. On the one hand, by the application of WKB analysis, we explore the asymptotic energy estimates of the wave equations influenced by four types of oscillating mechanisms. On the other hand, in order to prove the optimality of the energy estimates for the critical cases, typical coefficients and initial Cauchy data will be constructed to show the lower bound of the energy growth rate by the application of instability arguments.

Boundary control problem for a hyperbolic equation loaded along one of its characteristics

This paper investigates the unique solvability of the boundary control problem for a one-dimensional wave equation loaded along one of its characteristic curves in terms of a regular solution. The solution method is based on an analogue of the d’Alembert formula constructed for this equation. We point out that the domain of definition for the solution of DE, when the initial and final Cauchy data given on intervals of the same length is a square. The side of the squire is equal to the interval length. The boundary controls are established by the components of an analogue of the d’Alembert formula, which, in turn, are uniquely established by the initial and final Cauchy data. It should be noted that the normalized distribution and centering are employed in the final formulas of sought boundary controls, which is not typical for initial and boundary value problems initiated by equations of hyperbolic type.

Fundamental solution to 1D degenerate diffusion equation with locally bounded coefficients

In this work we study the degenerate diffusion equation ∂t=xαa(x)∂x2+b(x)∂x for (x,t)∈(0,∞)2, equipped with a Cauchy initial data and the Dirichlet boundary condition at 0. We assume that the order of degeneracy at 0 of the diffusion operator is α∈(0,2), and the coefficient functions (and their derivatives) are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution p(x,y,t) and prove several properties for p(x,y,t); by conducting a localization procedure, we obtain an approximation for p(x,y,t) for x,y in a neighborhood of 0 and t sufficiently small, where the error estimates only rely on the local bounds of a(x) and b(x) (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of α=1. Our work extends part of the existing results to the cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probabilistic view (e.g., wellposedness of stochastic differential equations).

On the Cauchy Problem for the Three-Dimensional Laplace Equation

In the present work, using the Carleman function, a harmonic function and its derivatives are restored by the Cauchy data on a part of the boundary of the region. It is shown that the effective construction of the Carleman function is equivalent to the construction of a regularized solution to the Cauchy problem. We assume that a solution to the problem exists and is continuously differentiable in a closed domain with precisely given Cauchy data. For this case, we establish an explicit formula for the continuation of the solution and its derivative, as well as the regularization formula for the case when under the indicated conditions instead of the initial Cauchy data their continuous approximations are given with a given error in the uniform metric. The stability estimates for the solution to the Cauchy problem in the classical sense are obtained.

The fundamental solution to 1D degenerate diffusion equation with one-sided boundary

In this work we adopt a combination of probabilistic approach and analytic method to study the fundamental solution to a certain type of one-dimensional degenerate diffusion equation. To be specific, we consider a diffusion equation on (0,∞) whose diffusion coefficient vanishes at the boundary 0, equipped with the Cauchy initial data and the Dirichlet boundary condition. One such diffusion equation that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0. Our main goal is to extend the study to cases when the diffusion coefficient has a general order of degeneracy, with a primary focus on the fundamental solution to such a degenerate diffusion equation. In particular, we study the regularity properties of the fundamental solution near 0, and investigate how the order of degeneracy of the diffusion operator and the Dirichlet boundary condition jointly affect these properties. We also provide estimates for the fundamental solution and its derivatives near 0.

Wide-angle mode parabolic equations for the modelling of horizontal refraction in underwater acoustics and their numerical solution on unbounded domains

The modelling of sound propagation in the ocean by the solution of mode parabolic equations is discussed. Mode parabolic equations can be obtained as the one-way approximation to horizontal refraction equations for modal amplitudes. Their wide-angle capabilities depend on the order of the Padé approximation of the involved pseudo-differential operators. Various aspects of numerical solution methods for wide-angle mode parabolic equations are considered in detail, including artificial domain truncation and Cauchy initial data for the point source field approximation. The capabilities of the discussed numerical approaches are demonstrated in several important test cases, including the problems of sound propagation in a penetrable wedge and in a sea with an underwater canyon.

A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.

Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$). In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordstrom Cauchy horizon. In the second paper of the series (arXiv:1702.05716), which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region

This is the second and last paper of a two-part series in which we prove the $$C^2$$ -formulation of the strong cosmic censorship conjecture for the Einstein–Maxwell–(real)–scalar–field system in spherical symmetry for two-ended asymptotically flat data. In the first paper, we showed that the maximal globally hyperbolic future development of an admissible asymptotially flat Cauchy initial data set is $$C^2$$ -future-inextendible provided that an $$L^2$$ -averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $$C^1$$ topology and is dense with respect to a (weighted) $$C^\infty $$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner–Nordstrom Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.

ЗАДАЧИ КОШИ В ГЕОМЕХАНИКЕ

Proposal about linear stress field of virgin solid and necessity for calculation stress-strain behavior near workings at actual mining cause to development analytical and numerical methods of the calculations. One-dimensional, two-dimensional and three-dimensional models of solids with relaxations, which are placed to the class of Couchy problems, for which Cauchy initial data are formulated, has been occurred. It is related to the fact that in rock mechanics plane with relaxation or space with cavity, for which there are infinitely remote points, are considered. There are known solutions, when boundary conditions represented by constants, determined by initial stress field adopted for concrete solid, are formulated on the infinity. In condition of numerical calculation, the software usually gives some result, accuracy of which does not control. Worldwide scientific schools represent same results which are in antimony with theory of equals of mathematical physics, which have outlined class of Cauchy problems for that with cancelled out boundary condition on the infinity. In the work, problems of rock mechanics for the plane relaxed with random holes are considered. The necessity to carry out points of the theory is proved. Method of solving of such problem class based on getting so-called additional solution formulated in class of Cauchy problem is proposed.

Functions represented into fractional Taylor series

Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.

Systems Parabolic in Petrovskii’s Sense in Hörmander Spaces

We study a general initial-boundary-value problem for systems parabolic in Petrovskii’s sense with trivial initial Cauchy data in some anisotropic Hormander inner-product spaces. It is proved that the operators corresponding to this problem are isomorphisms between the appropriate Hormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of the solutions of the problem. We also establish new sufficient conditions of continuity for the generalized partial derivatives of a given order for the chosen component of the solution.

On Classical Global Solutions of Nonlinear Wave Equations with Large Data

Abstract This article studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.

On an Inverse Problem for a One-Dimensional Two-Velocity Dynamical System

The evolution of the dynamical system under consideration is governed by the wave equation ρu tt − (γu x ) x + Au x + B u = 0, x>0, t > 0, with the zero initial Cauchy data and Dirichlet boundary control at x = 0. Here, ρ, γ, A, B are smooth 2 × 2–matrix-valued functions of x; ρ = diag {ρ1, ρ2} and γ = diag {γ1, γ2} are matrices with positive entries; u = u(x, t) is a solution (an ℝ2-valued function). In applications, the system corresponds to one-dimensional models, in which there are two types of wave modes, which propagate with different velocities and interact with each other. The “input→state” correspondence is realized by the response operator R : u(0, t) _→ γ(0)u x (0, t), t ≥ 0, which plays the role of inverse data. The representations for the coefficients A and B, which are used for their determination via the response operator, are derived. An example of two systems with the same response operator is given, where in the first system the wave modes do not interact, whereas in the second one the interaction does occur. Bibliography: 3 titles.

Construction of Cauchy data of vacuum Einstein field equations evolving to black holes

We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside. The proof makes use of the full strength of Christodoulou's work on the dynamical formation of black holes and Corvino-Schoen's work on the constructions of initial data set.

Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case

In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical La-grange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations. 1. Introduction. The method of quasi-reversibility has now a quite long history since the pioneering book of Latt es and Lions in 1967 [1]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter e. This perturbation has essentially two effects. Firstly the perturbation transforms the initial ill-posed problem into a well-posed one for any e, secondly the solution to such problem converges to the solution (if it exists) to the initial ill-posed problem when e tends to 0. Generally, the ill-posedness in the initial problem is due to unsuitable boundary conditions. As typical examples of linear ill-posed problems one may think of the backward heat equation, that is the initial condition is replaced by a final condition, or the heat or wave equations with lateral Cauchy data, that is the usual Dirichlet or Neumann boundary condition on the boundary of the domain is replaced by a pair of Dirichlet and Neumann boundary conditions on the same subpart of the boundary, no data being prescribed on the complementary part of the boundary.

On the spectrum of well-defined restrictions and extensions for the Laplace operator

The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator \(\hat L\) or a Volterra well-defined extension of a minimal operator L 0 generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator \(\hat L\) and of well-defined extensions of the minimal operator L 0 generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.

LONG TIME SOLUTIONS FOR WAVE MAPS WITH LARGE DATA

For (2 + 1)-dimensional wave maps with 𝕊2as the target, we show that for all positive numbers T0> 0 and E0> 0, there exist Cauchy initial data with energy at least E0, so that the solution's life-span is at least [0, T0]. We assume neither symmetry nor closeness to harmonic maps.

Space-time stationary solutions for the Burgers equation

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poissonian points. Moreover action-minimizing curves corresponding to different starting points merge with each other in finite time.