Global Uniqueness Theorem Research Articles

A model of liquid level measurements

A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level.

Global uniqueness theorems for linear and nonlinear waves

We prove a unique continuation from infinity theorem for regular waves of the form [ □ + V ( t , x ) ] ϕ = 0 . Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities, we show that the solution must vanish everywhere. The “no radiation” assumption is captured in a specific, finite rate of decay which in general depends on the L ∞ -profile of the potential V . We show that the result is optimal in many regards. These results are then extended to certain power-law type nonlinear wave equations, where the order of decay one must assume is independent of the size of the nonlinear term. These results are obtained using a new family of global Carleman-type estimates on the exterior of a null cone. A companion paper to this one explores further applications of these new estimates to such nonlinear waves.

A global uniqueness result for acoustic tomography of moving fluid

We consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in dimension d≥2. We give global uniqueness theorems for related inverse boundary value problem at fixed frequency.

A global existence and uniqueness result for a class of hyperbolic operators

In this paper a global existence and uniqueness theorem for the Cauchy problem related to the class of hyperbolic operators with double characteristics \(P=D^2_{t} - D^2_{x_1} - (t+ \lambda - \alpha (x_1))^2 D^2_{x_2},\) depending on the parameter \(\lambda \) in the half-space \(\Omega = \mathbb {R}^2 \times ]0, + \infty [\) is proved. In the first part of the paper, a priori estimates in Sobolev spaces \(W^{r,2}\), with \(r\) negative, have been established. Then, the regularity is studied by using these spaces and by constructing a Green’s function in the half-plane for the operator of the waves.

UNIQUENESS RESULTS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.

Coupled second order semilinear evolution equations indirectly damped via memory effects

In this paper, we study a system of coupled second order semilinear evolution equations in a Hilbert space, which is partially damped through memory effects. A global existence and uniqueness theorem regarding the solutions to its Cauchy problem is given. Following this, we analyze stability of the system energy, and obtain various decay rates corresponding to the integral kernel, some of them being optimal. Moreover, we apply our abstract theory to concrete systems, including that of Timoshenko type, for which we essentially improve the previous related results.

Global Solvability in Thermoelasticity with Second Sound on the Semi-axis

In this paper, we consider initial boundary value problem for the equations of one-dimensional nonlinear thermoelasticity with second sound in R+. First, we derive decay rates for linear systems which, in fact, is a hyperbolic systems with a damping term. Then, using this linear decay rates, we get L1 and L decay rates for nonlinear systems. Finally, combining with L2 estimates and a local existence theorem, we prove a global existence and uniqueness theorem for small smooth data. AMS Subject Classifications: 35L50, 74F05, 74H20 Chinese Library Classifications: O175.27

Functional equations on finite groups of substitutions

Motivated by some investigations of Babbage, we study a class of single variable functional equations. These are functional equations involving one unknown function and a finite set of known functions that form a group under the operation of composition. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. In the proof of the main result, the Implicit Function Theorem and, when the stabilizer is nontrivial, the Global Existence and Uniqueness Theorem play a key role.

A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Holmgren's theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwell's equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgren's Theorem.

A local inverse spectral theorem for Hamiltonian systems

We consider (2 × 2)-Hamiltonian systems of the form y′(x) = zJH(x)y(x), x ∊ [s−, s+). If a system of this form is in the limit point case, an analytic function is associated with it, namely its Titchmarsh–Weyl coefficient qH. The (global) uniqueness theorem due to de Branges says that the Hamiltonian H is (up to reparameterization) uniquely determined by the function qH. In this paper we give a local uniqueness theorem; if the Titchmarsh–Weyl coefficients and corresponding to two Hamiltonian systems are exponentially close, then the Hamiltonians H1 and H2 coincide (up to reparameterization) up to a certain point of their domain, which depends on the quantitative degree of exponential closeness of the Titchmarsh–Weyl coefficients.

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables. For this problem, we obtain sufficient conditions for the existence of generalized solutions. We separately consider the cases of many space variables and one space variable. We prove the global uniqueness theorem for a generalized solution without any time smallness conditions.

Functional equations and group substitutions

Motivated by some investigations of Babbage and a method of solving certain functional equations arising in competition problems, we investigate a class of functional equations and prove a local existence and uniqueness theorem for them. The main tools of the proof are the Inverse Function Theorem and the Global Existence and Uniqueness Theorem.

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

Symmetry results for overdetermined boundary value problems of nonlinear elliptic equations

Abstract We prove the radial symmetry of the solutions of second-order nonlinear elliptic equations for overdetermined Dirichlet and Neumann boundary value problems. In addition, a global uniqueness theorem of Holmgren type is given for nonlinear elliptic equations.

Local and Global Uniqueness Theorems of the N-th Order Differential Equations

Local and Global Uniqueness Theorems of the N-th Order Differential Equations

A moving-boundary problem for concrete carbonation: Global existence and uniqueness of weak solutions

This paper deals with a one-dimensional coupled system of semi-linear parabolic equations with a kinetic condition on the moving boundary. The latter furnishes the driving force for the moving boundary. The main result is a global existence and uniqueness theorem of positive weak solutions. The system under consideration is modelled on the so-called carbonation of concrete – a prototypical chemical-corrosion process in a porous solid – concrete – which incorporates slow diffusive transport, interfacial exchange between wet and dry parts of the pores and, in particular, a fast reaction in thin layers, here idealized as a moving-boundary surface in the solid. We include simulation results showing that the model captures the qualitative behaviour of the carbonation process.

On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives

Local and global existence and uniqueness theorems for a functional delay fractional differential equation with bounded delay are investigated. The continuity with respect to the initial function is proved under Lipschitz and the continuity kind conditions are analyzed.

Existence of the l-th moment of a solution to a stochastic functional-differential equation with the entire prehistory

The definition of a strong solution to a stochastic differential-functional equation with the entire prehistory is given, and basic inequalities required for obtaining existence and uniqueness theorems are proved. Global existence and uniqueness theorems are proved.