Carleman-type Estimate Research Articles

Unique continuation property for the Rosenau equation

In this work, using an appropriate Carleman-type estimate, we establish a unique continuation result for the Rosenau equation that models the dynamics of dense discrete systems with high order effects.

A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (eta ,Phi ) = (eta (x,y, t),Phi (x,y, t)) is a solution of the nonlinear system, in a suitable function space, and (eta ,Phi ) vanishes on an open subset Omega of mathbb {R}^2 times [-T,T], then (eta ,Phi )equiv 0 in the horizontal component of Omega . To state such property, we use a Carleman-type estimate for a differential operator mathcal {L} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.

Global Lipschitz stability of multiple space dependent ionic parameters for the MEA/hiPSC-CM assays

In this paper, we consider an inverse problem of determining two space dependent ionic parameters of a strongly coupled parabolic-elliptic reaction–diffusion system arising in cardiac electrophysiology modeling when simulating drugs action with multi-electrode array/human induced pluripotent stem cells-cardiomyocytes assays. We use the bidomain model coupled to an ordinary differential equation and we consider the classical phenomenological model in cardiac electrophysiology of FitzHugh–Nagumo to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate for two ionic parameters of the model using sub-boundary observations over an interval of time. The key ingredients are a global Carleman-type estimates with a suitable observations acting on a part of the boundary.

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d q>d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

On the asymptotic properties for stationary solutions to the Navier-Stokes equations

In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution \begin{document}$ v $\end{document} and its derivatives decay like \begin{document}$ O(|x|^{-k}) $\end{document} for sufficiently large \begin{document}$ k $\end{document} , depending on the velocity field, as \begin{document}$ |x|\to \infty $\end{document} , then \begin{document}$ v $\end{document} is zero on that exterior domain. Constructive estimate for \begin{document}$ k $\end{document} is given. In the case where velocity field is only bounded at infinity, we show that the infimum of \begin{document}$ L^2 $\end{document} norm of a velocity field on a unit ball located at distance \begin{document}$ t $\end{document} from an origin is bounded from below as \begin{document}$ Ce^{-\beta t^\frac 43\ln(t)}. $\end{document} The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.

Weighted gradient inequalities and unique continuation problems

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$\begin{aligned} \Vert e^{-\tau \ell (\cdot )} u^{\frac{1}{q}} f\Vert _q\le c_\tau \Vert e^{-\tau \ell (\cdot )} v^{\frac{1}{p}}\, \nabla f\Vert _p, \quad f\in C^\infty _0({\mathbb {R}}^n). \end{aligned}$$This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.

Quantitative uniqueness estimates for the generalized non-stationary Stokes system

ABSTRACTWe study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in with . One of our main results is a bound on the vanishing order of a non-trivial solution u satisfying the generalized non-stationary Stokes system, which is a quantitative version of the (strong) unique continuation property for u. Different from the previously known results, our unique continuation result only involves the velocity field u. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-cylinder inequalities for u.

On the cost of observability in small times for the one-dimensional heat equation

We aim at presenting a new estimate on the cost of observability in small times of the one-dimensional heat equation, which also provides a new proof of observability for the one-dimensional heat equation. Our proof combines several tools. First, it uses a Carleman-type estimate borrowed from our previous work (SIAM J. Control Optim. 56:3 (2018), 1692–1715), in which the weight function is derived from the heat kernel and which is therefore particularly easy. We also use explicit computations in the Fourier domain to compute the high-frequency part of the solution in terms of the observations. Finally, we use the Phragmen–Lindelof principle to estimate the low-frequency part of the solution. This last step is done carefully with precise estimations coming from conformal mappings.

Ionic parameters identification of an inverse problem of strongly coupled PDE’s system in cardiac electrophysiology using Carleman estimates

In this paper, we consider an inverse problem of determining multiple ionic parameters of a 2 × 2 strongly coupled parabolic–elliptic reaction–diffusion system arising in cardiac electrophysiology modeling. We use the bidomain model coupled to an ordinary differential equation (ODE) system and we consider a general formalism of physiologically detailed cellular membrane models to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate of the ion channels conductance parameters of the model using subboundary observations over an interval of time. The key ingredients are a global Carleman-type estimate with a suitable observations acting on a part of the boundary.

Stability of the unique continuation for the wave operator via Tataru inequality: the local case

In 1995, Tataru proved a Carleman-type estimate for linear operators with partially analytic coefficients that is generally used to prove the unique continuation of those operators. In this paper, we use this inequality to study the stability of the unique continuation in the case of the wave equation with coefficients independent of time. We prove a logarithmic estimate in a ball whose radius has an explicit dependence on the C1-norm of the coefficients and on the other geometric properties of the operator.

Non-existence of time-periodic vacuum space-times

We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in [Bicak, Scholtz, and Tod; 2010]. The proof relies on extending a suitably constructed "candidate" Killing vector field from null infinity, via Carleman-type estimates obtained in [Alexakis, Schlue, Shao; 2013].

A Proof of Friedman’s Ergosphere Instability for Scalar Waves

Let $(\mathcal{M}^{3+1},g)$ be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $\mathscr{E}$ and no future event horizon $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic argument that the energy of certain solutions $\phi$ of $\square_{g}\phi=0$ grows to $+\infty$ as time increases. In this paper, we provide a rigorous proof of Friedman's instability. Our setting is, in fact, more general. We consider smooth spacetimes $(\mathcal{M}^{d+1},g)$, for any $d\ge2$, not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $\partial\mathscr{E}$ of $\mathscr{E}$ on a small neighborhood of a point $p\in\partial\mathscr{E}$. This condition always holds if $(\mathcal{M},g)$ is analytic in that neighborhood of $p$, but it can also be inferred in the case when $(\mathcal{M},g)$ possesses a second Killing field $\Phi$ such that the span of $\Phi$ and the stationary Killing field $T$ is timelike on $\partial\mathscr{E}$. We also allow the spacetimes $(\mathcal{M},g)$ under consideration to possess a (possibly empty) future event horizon $\mathcal{H}^{+}$, such that, however, $\mathcal{H}^{+}\cap\mathscr{E}=\emptyset$ (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira, Cardoso and Crispino. Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes.

A weighted identity for stochastic partial differential operators and its applications

In this paper, a weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in establishing Carleman-type estimates for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce some known global Carleman estimates for stochastic parabolic equations, stochastic Schrödinger equations, stochastic transport equations and their deterministic counterparts. Meanwhile, as its applications, we derive two different Carleman estimates for linear forward stochastic complex Ginzburg–Landau equations. They can be used to study the controllability/observability and inverse problems for some stochastic complex Ginzburg–Landau equations, respectively.

Stability results for the parameter identification inverse problem in cardiac electrophysiology

In this paper we prove a stability estimate of the parameter identification problem in cardiac electrophysiology modeling. We use the monodomain model which is a reaction diffusion parabolic equation where the reaction term is obtained by solving an ordinary differential equation (ODE). We are interested in proving the stability of the identification of the parameter , which is the parameter that multiplies the cubic term in the reaction term. The proof of the result is based on a new Carleman-type estimate for both partial differential equation (PDE) and ODE problems. As a consequence of the stability result we prove the uniqueness of the parameter giving some observations of both state variables at a given time t0 in the whole domain and in the PDE variable in a non empty open subset w0 of the domain.

On the Reachable Set for the One-Dimensional Heat Equation

The goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x $\in$ (--L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 \textgreater{} L any function which can be extended analytically on the square {x + iy, |x| + |y| $\le$ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| \textless{} L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.

Stability of the unique continuation for the wave operator via Tataru inequality and applications

In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in an arbitrary domain of Rn+1, where all the parameters are calculated explicitly in terms of the C1-norm of the coefficients and on the other geometric properties of the problem. We use the Carleman-type estimate proved by Tataru in 1995 and an iteration of the local stability. We apply the result to the case of a wave equation with data on a cylinder and we get a stable estimate for any positive time, also after the first conjugate point associated with the geodesics of the metric of the variable coefficients.

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.

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part II: an inverse source problem

In this paper, we consider Carleman-type estimate and consider an inverse problem for second order hyperbolic systems in an anisotropic case. In the previous Part I paper, we established a Carleman-type estimate for hyperbolic systems in which the coefficient matrices satisfy suitable conditions. We apply a Carleman estimate in the previous Part I paper to an inverse source problem for second-order hyperbolic systems in an anisotropic case and prove an estimate of the Hölder type.

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates

In this paper, we consider Carleman-type estimates and an inverse source problem for second-order hyperbolic systems in an anisotropic case. In Part I, we establish a Carleman-type estimate for hyperbolic systems in an anisotropic case by assuming that the coefficient matrices in the systems satisfy suitable conditions and taking the weight function where and for in a bounded domain in with -boundary and .