Search results for "uniqueness"

On a nonlinear flux-limited equation arising in the transport of morphogens

2012

Abstract Motivated by a mathematical model for the transport of morphogens in biological systems, we study existence and uniqueness of entropy solutions for a mixed initial–boundary value problem associated with a nonlinear flux-limited diffusion system. From a mathematical point of view the problem behaves more as a hyperbolic system than a parabolic one.

Existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets

2005

A quasilinear parabolic equation with quadratic gradient terms is analyzed. The equation models an optimal portfolio in so-called incomplete financial markets consisting of risky assets and non-tradable state variables. Its solution allows to compute an optimal portfolio strategy. The quadratic gradient terms are essentially connected to the assumption that the so-called relative risk aversion function is not logarithmic. The existence of weak global-in-time solutions in any dimension is shown under natural hypotheses. The proof is based on the monotonicity method of Frehse. Furthermore, the uniqueness of solutions is shown under a smallness condition on the derivatives of the covariance (?…

Non-Lipschitz Homogeneous Volterra Integral Equations

2018

In this chapter we introduce a class of nonlinear Volterra integral equations (VIEs) which have certain properties that deviate from the standard results in the field of integral equations. Such equations arise from various problems in shock wave propagation with nonlinear flux conditions. The basic equation we will consider is the nonlinear homogeneous Hammerstein–Volterra integral equation of convolution type $$\displaystyle u(t) = \int _0^t k(t-s) g(u(s))\,\mathrm {d}s. $$ When g(0) = 0, this equation has function u ≡ 0 as a solution (trivial solution). It is interesting to determine whether there exists a nontrivial solution or not. Classical results on integral equations are not to be …

On the condition number of the antireflective transform

2010

Abstract Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for …

A note on the uniqueness and attractive behavior of solutions for nonlinear Volterra equations

2001

In this paper we prove that positive solutions of some nonlinear Volterra integral equations must be locally bounded and global attractors of positive functions. These results complete previous results about the existence and uniqueness of solutions and their attractive behavior.

Convergence of a finite volume scheme for the compressible Navier–Stokes system

2019

We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.

Well-posedness of the boundary layer equations

2004

We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. Theproof is achieved applying the abstract Cauchy--Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433--461], as we do not require analyticity of the data with respect to the normal variable.

Existence and uniqueness for the Prandtl equations

2001

International audience; Under the hypothesis of analyticity of the data with respect to the tangential variable we prove the existence and uniqueness of the mild solution of Prandtl boundary layer equation. This can be considered an improvement of the results of [8] as we do not require analyticity with respect to the normal variable. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.

2015

In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrodinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain S…

On the Uniqueness of the Energy and Momenta of an Asymptotically Minkowskian Space-Time: The Case of the Schwarzschild Metric

2013

Some theorems about the uniqueness of the energy of asymptotically Minkowskian spaces are recalled. The suitability of almost everywhere Gauss coordinates to define some kind of physical energy in these spaces is commented. Schwarzschild metric, when its source radius is larger than the Schwarzschild radius and in the case of a black hole, is considered. In both cases, by using a specific almost everywhere Gaussian coordinate system, a vanishing energy results. We explain why this result is not in contradiction with the quoted theorems. Finally we conclude that this metric is a particular case of what we have called elsewhere a creatable universe.