Carleman Type Inequality Research Articles

Backward Uniqueness for 3D Navier–Stokes Equations With Non-Trivial Final Data and Applications

Abstract Presented is a backward uniqueness result of bounded mild solutions of 3D Navier–Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate that enables to treat terms involving Calderon–Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

An inverse source problem for the Schrödinger equation with variable coefficients by data on flat subboundary

We consider an inverse source problem for the Schrödinger equation with variable coefficients. We prove the uniqueness of solution of the problem by data on a flat subboundary over time interval, under a certain condition of the coefficients of the principal terms. We first reduce the inverse problem to a Cauchy problem for a system of integro-differential equations by using Fourier transform. Next, we establish a pointwise Carleman type inequality which is the key tool in the proof of our main result.

Quantitative dependence of some discrete limiting classes on the Muckenhoupt \calA 1 ( u ) class

UDC 517.5 We prove some relations between the discrete Gehring classes 𝒢 q and the discrete Muckenhoupt classes 𝒜 p . Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between 𝒢 1 , 𝒜 ∞ and 𝒜 1 for nonincreasing and nondecreasing weights. Finally, we establish some general results by introducing the notions of 𝒢 φ classes defined for nonnegative convex function φ .

Carleman type inequalities for fractional relativistic operators

In this paper we derive Carleman estimates for the fractional relativistic operator. We consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals to deduce Carleman estimates with linear exponential weight. Our approach is based on spectral methods and functional calculus.

Uniqueness of solution of an inverse source problem for ultrahyperbolic equations

The aim of this article is to investigate the uniqueness of solution of an inverse source problem for ultrahyperbolic equations. We first reduce the inverse problem to a Cauchy problem for an integro-differential equation and then by using a pointwise Carleman type inequality we prove the uniqueness.

An inequality involving the constant e and a generalized Carleman-type inequality

An inequality involving the constant e and a generalized Carleman-type inequality

Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we state the stability results for our inverse coefficient problem. Secondly, we establish and prove two Carleman type inequalities both for the solutions and for the unknown coefficients. Finally, we complete the proofs of the stability results in terms of the above Carleman estimates.

Approximation formulas for the constant e and an improvement to a Carleman-type inequality

We give an explicit formula for the determination of the coefficients cj appearing in the expansionx(1+∑j=1qcjxj)(πΓ(x+12))1/x=e+O(1xq+1) for x→∞ and q∈N:={1,2,…}. We also derive a pair of recurrence relations for the determination of the constants λℓ and μℓ in the expansion(1+1x)x∼e(1+∑ℓ=1∞λℓ(x+μℓ)2ℓ−1) as x→∞. Based on this expansion, we establish an inequality for (1+1/x)x. As an application, we give an improvement to a Carleman-type inequality.

On Some Riesz and Carleman Type Inequalities for Harmonic Functions in the Unit Disk

We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $$h^p$$ , $$p>1$$ , and for complex harmonic functions in $$h^4$$ . The results extend some recent results in the area. Further, we discuss some Riesz type results for holomorphic functions.

Pad\xe9 approximant related to inequalities involving the constant e and a generalized Carleman-type inequality

Based on the Padé approximation method, in this paper we determine the coefficients a_{j} and b_{j} (1leq j leq k) such that \t\t\t1e(1+1x)x=xk+a1xk−1+⋯+akxk+b1xk−1+⋯+bk+O(1x2k+1),x→∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\frac{1}{e} \\biggl( 1+\\frac{1}{x} \\biggr) ^{x}= \\frac{x^{k}+a_{1}x^{k-1}+ \\cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\\cdots +b_{k}}+O \\biggl( \\frac{1}{x ^{2k+1}} \\biggr) , \\quad x\\to \\infty , $$\\end{document} where kgeq 1 is any given integer. Based on the obtained result, we establish new upper bounds for ( 1+1/x ) ^{x}. As an application, we give a generalized Carleman-type inequality.

On the profile of energy concentration at blow-up points for subconformal focusing nonlinear waves

We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the H 1 H^1 -norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this H 1 H^1 -energy can be distributed inside such cones. Our proof relies on new localized estimates—obtained using Carleman-type inequalities—for such nonlinear waves. These bound the L p + 1 L^{p+1} -norm in the interior of timelike cones by their H 1 H^1 -norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones.

Null controllability of a population dynamics with degenerate diffusion

In this paper, we are interested in the null controllability of a linear population dynamics model with degenerate dispersion coefficient. We develop first a Carleman-type inequality for its adjoint system, and then an observability inequality which allows us to establish the existence of a control acting on a subset of the space domain which steers the population of a certain age to extinction in a finite time.

Quantitative uniqueness for Schrödinger operator with regular potentials

We give a sharp upper bound on the vanishing order of solutions to the Schrödinger equation with electric and magnetic potentials on a compact smooth manifold. Our main result is that the vanishing order of nontrivial solutions to Δu + V · ∇ u + Wu = 0 is everywhere less than . Our method is based on quantitative Carleman type inequalities, and it allows us to show the following uniform doubling inequality urn:x-wiley:01704214:media:mma2951:mma2951-math-0003 which implies the desired result. Copyright © 2013 John Wiley & Sons, Ltd.

A Carleman Type Inequality for Sugeno Integrals

One of the famous mathematical inequalities is Carlemans inequality. It is an important inequality from both mathematical and application points of view. In this paper, a Carleman type inequality for Sugeno integrals is studied.

Strong Unique Continuation for the Navier–Stokes Equation with Non-Analytic Forcing

We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator.

Quantitative uniqueness for Schroedinger operator

We give an upper bound on the vanishing order of solutions to Schrodinger equation on a compact smooth manifold. Our method is based on Carleman type inequalities, and gives a generalisation to a result of H. Donnely and C. Fefferman [DF88] on eigenfunctions. It also sharpens previous results of I. Kukavica [Kuk98].

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.

Controllability of a Reaction-Diffusion System Describing Predator–Prey Model

This article establishes the controllability to the trajectories of a reaction-diffusion-advection system describing predator–prey model by using distributed controls acting on a single equation with the no-flux boundary conditions. We first prove the exact null controllability of an associated linearized problem by establishing an observability estimate, derived from a global Carleman type inequality, for the adjoint system. The proof of the nonlinear problem relies on the suitable regularity of the control and Kakutani's fixed point theorem.

Exact null controllability of a semilinear parabolic equation arising in finance

This paper is concerned with the exact null controllability of certain semilinear Black–Scholes type equations in a bounded interval of R + with Neumann boundary conditions. The control is assumed to be distributed along a subset ω of I . First, we prove the exact null controllability of an associated linearized problem by establishing the observability estimate derived from a Carleman type inequality which is the key result for the whole theory. Then, the exact null controllability of the nonlinear problem is discussed using the infinite-dimensional Kakutani fixed point theorem.

On a class of non-linear parabolic control systems with memory effects

This work establishes the local exact null controllability of non-linear parabolic integrodifferential equations with non-linearities under the integral sign in a bounded domain and with homogeneous Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain ω of Ω. Using a Carleman type inequality we first prove the global exact null controllability of a linearized equation. Then, by a fixed point method, we obtain the main result for the non-linear case. This result asserts that, when the non-linearity is of class C 1 and globally Lipschitz, the system is null controllable.