arXiv subjects

Leander Claes

Publications and source records attributed to Leander Claes.

4 recordsLinked to original sources

arXiv2026

Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

This manuscript is concerned with the evolution system \[ \left\{ \begin{array}{l} u_{ttt} + \alpha u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + \big( \widehat{\gamma}(\Theta) u_x\big)_x, \Theta_t = D \Theta_{xx} + \Gamma(\Theta) u_{xt}^2, \end{array} \right. \] which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type. Under the assumptions that $D>0$ and $\alpha\ge 0$, and that $\gamma, \widehat{\gamma}$ and $\Gamma$ are sufficiently smooth with $\gamma>0, \widehat{\gamma}>0$ and $\Gamma\ge 0$ on $[0,\infty)$, for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.

math.AP↗

arXiv2025

Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by $W^{1,p}$ energy analysis

In bounded $n$-dimensonal domains with $n\ge 1$, this manuscript examines an initial-boundary value problem for the system \[ \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma(\Theta) \nabla u_t) + a \nabla \cdot (\gamma(\Theta) \nabla u) + \nabla\cdot f(\Theta), \Theta_t = D\Delta\Theta + \Gamma(\Theta) |\nabla u_t|^2 + F(\Theta)\cdot \nabla u_t, \end{array} \right. \] which in the case $n=1$ and with $\gamma\equiv \Gamma$ as well as $f\equiv F$ reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides $f$ and $F$, also the core ingredients $\gamma$ and $\Gamma$ may depend on the temperature variable $\Theta$. Firstly, a statement on local existence of classical solutions is derived for arbitrary $a>0, D>0$ as well as $0<\gamma\in C^2([0,\infty))$ and $0\le\Gamma\in C^1([0,\infty))$, for functions $f\in C^2([0,\infty);{\mathbb{R}}^n)$ and $F\in C^1([0,\infty);{\mathbb{R}}^n)$ with $F(0)=0$, and for suitably regular initial data of arbitrary size. Secondly, it is seen that for each $p\ge 2$ such that $p>n$ there exists $\delta(p)>0$ with the property that whenever in addition to the above we have \[ \frac{a}{\gamma(0)} \le \delta(p) \qquad \mbox{and} \qquad \frac{|f'(\Theta_\star)| \cdot |F(\Theta_\star)|}{D \cdot \gamma(\Theta_\star)} \le \delta(p), \] for initial data suitably close to the constant level given by $u=0$ and $\Theta=\Theta_\star$, with any fixed $\Theta_\star\ge 0$, these solutions are actually global in time and have the property that $\nabla u_t, \nabla u$ and $\nabla\Theta$ decay exponentially fast in $L^p$. This is achieved by detecting suitable dissipative properties of functionals involving norms of these gradients in $L^p$ spaces.

math.AP↗

arXiv2024

A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions

A model for the generation of heat due to mechanical losses during acoustic wave propagation in a solid is considered in a Kelvin-Voigt type framework. In contrast to previous studies on related thermoviscoelastic models, in line with recent experimental findings the present manuscript focuses on situations in which elastic parameters depend on temperature. Despite an apparent loss of mathematically favorable structural properties thereby encountered, in the framework of a suitably generalized concept of solvability a result on global existence of solutions is derived under mild assumptions which, in particular, do not involve any smallness condition on the initial data.

math.AP↗

arXiv2020

Estimation of acoustic wave non-linearity in ultrasonic measurement systems

Most measurement methods based on ultrasound, such as sound velocity, absorption or flow measurement systems, require that the acoustic wave propagation is linear. In many cases, linear wave propagation is assumed due to small signal amplitudes or verified, for example, by analysing the received signal spectra for the generation of harmonic frequency components. In this contribution, we present an approach to quantify occurrence of non-linear effects of acoustic wave propagation in ultrasonic measurement systems based on the evaluation of the acoustic Reynolds number. One parameter required for the determination of the acoustic Reynolds number is the particle velocity of the acoustic wave, which is not trivially obtained in most measurement systems. We thus present a model-based approach to estimate the particle velocity of an acoustic wave by identifying a Mason model from electrical impedance measurements of a given transducer. The Mason model is then used to determine the transducer's velocity output for a given electrical signal, allowing for an evaluation of the acoustic Reynolds number for different target media.

physics.flu-dyn↗