Weak observability estimates for 1-D wave equations with rough coefficients

DOI10.1016/j.anihpc.2013.10.004zbMath1320.93027arXiv1305.0530OpenAlexW2963084856MaRDI QIDQ2347862

Francesco Fanelli, Enrique Zuazua

Publication date: 10 June 2015

Published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1305.0530

zbMATH Keywords

loss of derivativesobservability estimatesnon-Lipschitz coefficientsZygmund regularity1-D wave equation

Mathematics Subject Classification ID

Observability (93B07) Wave equation (35L05) PDEs in connection with control and optimization (35Q93)

Related Items (12)

A Few Remarks on Hyperbolic Systems with Zygmund in Time Coefficients ⋮ Observability estimates for the wave equation with rough coefficients ⋮ Sidewise profile control of 1-D waves ⋮ The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients ⋮ On the cost of observability in small times for the one-dimensional heat equation ⋮ Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach ⋮ On the Cauchy problem for microlocally symmetrizable hyperbolic systems with log-Lipschitz coefficients ⋮ Semi-Classical Observation Suffices for Observability: Wave and Schrödinger Equations ⋮ Some local questions for hyperbolic systems with non-regular time dependent coefficients ⋮ Embeddings and characterizations of Lipschitz spaces ⋮ Low regularity ray tracing for wave equations with Gaussian beams ⋮ Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

Cites Work

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• A well-posedness result for hyperbolic operators with Zygmund coefficients
• Sharp counterexamples for Strichartz estimates for low regularity metrics
• Transport equation and Cauchy problem for BV vector fields
• Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem
• Sharp \(L^2\to L^q\) bounds on spectral projectors for low regularity metrics
• Para-differential calculus and applications to the Cauchy problem for nonlinear systems
• On the well-posedness of the incompressible density-dependent Euler equations in the \(L^p\) framework
• Functional analysis, Sobolev spaces and partial differential equations
• Concerning the \(L^ p\) norm of spectral clusters for second-order elliptic operators on compact manifolds
• Exact controllability for semilinear wave equations in one space dimension
• Uniqueness of continuous solutions for BV vector fields
• Concentration and lack of observability of waves in highly heterogeneous media
• Hyperbolic operators with non-Lipschitz ceofficients
• On Strichartz and eigenfunction estimates for low regularity metrics
• A generalized interpolation inequality and its application to the stabilization of damped equations
• On the null controllability of the one-dimensional heat equation with BV coefficients
• On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials
• Wave propagation, observation and control in 1-\(d\) flexible multi-structures.
• Addendum to ``Concentration and lack of observability of waves in highly heterogeneous media
• Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients, II
• Fourier Analysis and Nonlinear Partial Differential Equations
• Spectral cluster estimates for C 1,1 metrics
• Some examples of hyperbolic equations without local solvability
• Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time
• Stabilization of the Wave Equation on 1-d Networks
• The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations
• Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
• Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary
• The rate at which energy decays in a string damped at one end
• Dependence of High-Frequency Waves with Respect to Potentials
• Singular Optimal Control for a Transport-Diffusion Equation
• Questions of Existence and Uniqueness for Hyperbolic Equations with Discontinuous Coefficients
• Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

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