arXiv subjects

Anna Rozanova-Pierrat

Publications and source records attributed to Anna Rozanova-Pierrat.

23 records · Page 2Linked to original sources

arXiv2018

Cauchy Problem for the Kuznetsov Equation

We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the non-viscous case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2-stability estimate, obtained between a regular and a less regular solutions.

math.AP↗

arXiv2017

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for $n$ and $d$-sets.

math.FA↗

arXiv2016

Approximation of a compressible Navier-Stokes system by non-linear acoustical models

We analyse the existing derivation of the models of non-linear acoustics such as the Kuznetsov equation, the NPE equation and the KZK equation. The technique of introducing a corrector in the derivation ansatz allows to consider the solutions of these equations as approximations of the solution of the initial system (a com-pressible Navier-Stokes/Euler system). The validation of the approximation ansatz is given for the KZK equation case.

math.AP↗

arXiv2015

Short time heat diffusion in compact domains with discontinuous transmission boundary conditions

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(ε,δ)$-domains $Ω\subset\mathbb{R}^n$ with a $d$-set boundary (for instance, aself-similar fractal), we find the first term of the small-timeasymptotic expansion of the heat content in the complement of$Ω$, and also the second-order term in the case of a regularboundary. The asymptotic expansion is different for the cases offinite and infinite resistance of the boundary. The derived formulasrelate the heat content to the volume of the interior Minkowskisausage and present a mathematical justification to the de Gennes'approach. The accuracy of the analytical results is illustrated bysolving the heat problem on prefractal domains by a finite elementsmethod.

math.AP↗

arXiv2010

Perturbative numeric approach in microwave imaging

In this paper, we show that using measurements for different frequencies, and using ultrasound localized perturbations it is possible to extend the method of the imaging by elastic deformation developed by Ammari and al. [Electrical Impedance Tomography by Elastic Deformation SIAM J. Appl. Math., 68(6), (2008), 1557-1573.] to problems for the Helmholtz equations with Neumann boundary conditions, and to reconstruct by a perturbation method both the conductivity and the permittivity, provided that the conductivity function is coercive and the wave number in the Helmholtz equation is not a resonant frequency.

math.AP↗