CARLEMAN ESTIMATE AND UNIQUE CONTINUATION PROPERTY FOR THE LINEAR STOCHASTIC KORTEWEG–DE VRIES EQUATION

2014 ◽  
Vol 90 (2) ◽  
pp. 283-294 ◽  
Author(s):  
PENG GAO
Keyword(s):  
Galerkin Method ◽  
Uniqueness Theorem ◽  
Vries Equation ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Well Posedness ◽  
Unique Continuation Property ◽  
De Vries ◽  
Continuation Property ◽  
The Galerkin Method

AbstractIn this paper, we obtain the well posedness of the linear stochastic Korteweg–de Vries equation by the Galerkin method, and then establish the Carleman estimate, leading to the unique continuation property (UCP) for the linear stochastic Korteweg–de Vries equation. This UCP cannot be obtained from the classical Holmgren uniqueness theorem.

A NEW UNIQUE CONTINUATION PROPERTY FOR THE KORTEWEG–DE VRIES EQUATION

2014 ◽  
Vol 90 (1) ◽  
pp. 90-98 ◽  
Author(s):  
MO CHEN ◽  
PENG GAO
Keyword(s):  
Vries Equation ◽  
Finite Interval ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
De Vries ◽  
Continuation Property ◽  
Global Carleman Estimate

AbstractThe aim of this paper is to obtain a new unique continuation property (UCP) for the Korteweg–de Vries equation posed on a finite interval. Compared with the previous UCP, we need fewer conditions on the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg–de Vries equation.

Local state observation for stochastic hyperbolic equations

2020 ◽  
Vol 26 ◽  
pp. 79
Author(s):  
Qi Lü ◽  
Zhongqi Yin
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equations ◽  
Unique Continuation ◽  
Local State ◽  
Carleman Estimate ◽  
State Observation ◽  
Unique Continuation Property ◽  
Continuation Property ◽  
Global Carleman Estimate

In this paper, we solve a local state observation problem for stochastic hyperbolic equations without boundary conditions, which is reduced to a local unique continuation property for these equations. This result is proved by a global Carleman estimate. As far as we know, this is the first result in this topic.

scholarly journals Unique continuation property of solutions to general second order elliptic systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Naofumi Honda ◽  
Ching-Lung Lin ◽  
Gen Nakamura ◽  
Satoshi Sasayama
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Unique Continuation ◽  
Elliptic Systems ◽  
General System ◽  
Second Order ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
Constant Coefficients ◽  
Continuation Property

Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

2018 ◽  
Vol 149 (1) ◽  
pp. 219-260 ◽  
Author(s):  
F. A. Gallego ◽  
A. F. Pazoto
Keyword(s):  
Burgers Equation ◽  
Unique Continuation ◽  
Global Solutions ◽  
Exponential Stabilization ◽  
Well Posedness ◽  
Interpolation Theory ◽  
Damping Term ◽  
Unique Continuation Property ◽  
De Vries ◽  
Compactness Arguments

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

scholarly journals Discrete Carleman estimates and three balls inequalities

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Aingeru Fernández-Bertolin ◽  
Luz Roncal ◽  
Angkana Rüland ◽  
Diana Stan
Keyword(s):  
Unique Continuation ◽  
Carleman Estimate ◽  
Carleman Estimates ◽  
Auxiliary Result ◽  
Logarithmic Convexity ◽  
Unique Continuation Property ◽  
Discrete Operators ◽  
Continuation Property ◽  
Discrete Setting ◽  
The Continuum

AbstractWe prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.

scholarly journals Optimal three-ball inequality, quantitative uniqueness for the bi-Laplace equations

2020 ◽  
Author(s):  
Xiaoyu FU ◽  
Zhonghua LIAO
Keyword(s):  
Laplace Equation ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Quantitative Version ◽  
Laplace Equations ◽  
Connected Set ◽  
Unique Continuation Property ◽  
Continuation Property

In this paper, we prove an optimal three-ball inequality for bi-Laplace equation in some open, connected set. The derivation of such estimate relies on a delicate Carleman estimate for the bi-Laplace equation and some Caccioppoli inequalities to estimate the lower-ters. Based on three -ball inequality, we then derive the vanishing order of solutions, which is a quantitative version of the strong unique continuation property.

scholarly journals Strict monotonicity and unique continuation of the biharmonic operator - doi: 10.5269/bspm.v30i2.14502

1970 ◽  
Vol 30 (2) ◽  
pp. 79-83
Author(s):  
Najib Tsouli ◽  
Omar Chakrone ◽  
Mostafa Rahmani ◽  
Omar Darhouche
Keyword(s):  
Unique Continuation ◽  
Biharmonic Operator ◽  
Strict Monotonicity ◽  
Unique Continuation Property ◽  
Continuation Property

In this paper, we will show that the strict monotonicity of the eigenvalues of the biharmonic operator holds if and only if some unique continuation property is satisfied by the corresponding eigenfunctions.

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