THE MINIMIZING TOTAL VARIATION FLOW WITH MEASURE INITIAL CONDITIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 431-494 ◽  
Author(s):  
F. ANDREU ◽  
J. M. MAZÓN ◽  
J. S. MOLL ◽  
V. CASELLES
Keyword(s):  
Total Variation ◽  
Initial Conditions ◽  
Singular Part ◽  
Dimensional Manifold ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
Limit Solution ◽  
Total Variation Flow ◽  
The Cauchy Problem ◽  
Limit Solutions

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝN. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L1(ℝN). We are able to characterize limit solutions when the initial condition μ=h+μs, where h∈L1(ℝN)∩L∞(ℝN), and μs=αℋk⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k<N-1 we prove that the singular part of the solution does not move, it remains equal to μs for all t≥0. In particular, u(t)=δ0 when u(0)=δ0. In case k=N-1 we prove that the singular part of the limit solution is [Formula: see text] and we also characterize its absolutely continuous part. This explicit behaviour permits to characterize limit solutions. We also give an entropy condition characterization of the solution which is more satisfactory when k<N-1. Finally, we describe some distributional solutions which do not have the behaviour characteristic of limit solutions.

scholarly journals Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls

2019 ◽  
Vol 24 (2) ◽  
pp. 189-209
Author(s):  
Yatian Pei ◽  
Yong-Kui Chang
Keyword(s):  
Control System ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Optimal Controls ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
Nonlocal Initial Condition

In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.

scholarly journals The uniqueness of bounded or measurable solutions of some functional equations

1970 ◽  
Vol 11 (2) ◽  
pp. 186-190
Author(s):  
T. D. Howroyd
Keyword(s):  
Banach Algebra ◽  
Measurable Function ◽  
Functional Equations ◽  
Positive Measure ◽  
Continuous Solution ◽  
Initial Conditions ◽  
Initial Condition ◽  
Real Solution ◽  
Uniqueness Of Solutions ◽  
Image Position

In this paper we are concerned with the uniqueness of solutions of functional equations of the form Some conditions for (1) or (2) to have at most one real continuous solution f which satisfies two given initial conditions are contained in [2], [3], [4] and [7]. Conditions sufficient for the equation to determine at most one continuous solution f with values in a Banach algebra are contained in [5]. It is well known (see [1] ch. 2) that one initial condition suffices for Cauchy's equation or two for Jensen's equation to uniquely determine a real solution f which is bounded on an interval or majorized on a set of positive measure by a measurable function. We place conditions on F and H so that similar statements can be made about solutions of (1) or (2). The corresponding results for solutions which are functions of many real variables follow as for Cauchy's and Jensen's equations (see [1] ch. 5).

scholarly journals Construction of a solution of a certain evolution equation

1977 ◽  
Vol 66 ◽  
pp. 23-36 ◽  
Author(s):  
Akinobu Shimizu
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Differential Operator ◽  
Evolution Equation ◽  
Partial Differential Operator ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
The Cauchy Problem

Let us consider the stochastic differential equation,with initial condition,where Bt, t ≧ 0,is a one-dimensional Brownian motion, and Lxis a second order uniformly elliptic partial differential operator satisfying some additional conditions that will be described in §2. The existence and the uniqueness of solutions of the Cauchy problem have been established by B. L. Rozovskii [8].

Is it a Right Assumption That B and Ge are Distributed Randomly After Growing a Strained HBT-Structure?

2002 ◽  
Vol 716 ◽  
Author(s):  
Victor I. Kol'dyaev
Keyword(s):  
Surface Segregation ◽  
Initial Conditions ◽  
Initial Condition ◽  
Basic Assumptions ◽  
Transient Diffusion ◽  
Segregation Process ◽  
Main Observation

AbstractIt is accepted that surface Ge atoms are considered to be responsible for the surface B segregation process. A set of original experiments is carried out. A main observation from the B and Ge profiles grown at different conditions shows that at certain conditions B is taking initiative and determine the Ge surface segregation process. basic assumptions are suggested to self-consistently explain these original experimental features and what is observed in the literature. These results have a strong implication for modeling the B diffusion in Si1-xGex where the initial conditions should be formulated accounting for the correlation in B and Ge distribution. A new assumption for the initial condition to be “all B atoms are captured by Ge” is regarded as a right one implicating that there is no any transient diffusion representing the B capturing kinetics.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

scholarly journals Application of Adjoint-Derived Forecast Sensitivities to the 24–25 January 2000 U.S. East Coast Snowstorm

2005 ◽  
Vol 133 (11) ◽  
pp. 3148-3175 ◽  
Author(s):  
Daryl T. Kleist ◽  
Michael C. Morgan
Keyword(s):  
Response Function ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Lower Troposphere ◽  
Vertical Motion ◽  
Initial Condition ◽  
Response Functions ◽  
Eastern United States ◽  
Surface Cyclone

Abstract The 24–25 January 2000 eastern United States snowstorm was noteworthy as operational numerical weather prediction (NWP) guidance was poor for lead times as short as 36 h. Despite improvements in the forecast of the surface cyclone position and intensity at 1200 UTC 25 January 2000 with decreasing lead time, NWP guidance placed the westward extent of the midtropospheric, frontogenetically forced precipitation shield too far to the east. To assess the influence of initial condition uncertainties on the forecast of this event, an adjoint model is used to evaluate forecast sensitivities for 36- and 48-h forecasts valid at 1200 UTC 25 January 2000 using as response functions the energy-weighted forecast error, lower-tropospheric circulation about a box surrounding the surface cyclone, 750-hPa frontogenesis, and vertical motion. The sensitivities with respect to the initial conditions for these response functions are in general very similar: geographically isolated, maximized in the middle and lower troposphere, and possessing an upshear vertical tilt. The sensitivities are maximized in a region of enhanced low-level baroclinicity in the vicinity of the surface cyclone’s precursor upper trough. However, differences in the phase and structure of the gradients for the four response functions are evident, which suggests that perturbations could be constructed to alter one response function but not necessarily the others. Gradients of the forecast error response function with respect to the initial conditions are used in an iterative procedure to construct initial condition perturbations that reduce the forecast error. These initial condition perturbations were small in terms of both spatial scale and magnitude. Those initial condition perturbations that were confined primarily to the midtroposphere grew rapidly into much larger amplitude upper-and-lower tropospheric perturbations. The perturbed forecasts were not only characterized by reduced final time forecast error, but also had a synoptic evolution that more closely followed analyses and observations.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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