arXiv subjects

Stéphane Junca

Publications and source records attributed to Stéphane Junca.

16 recordsLinked to original sources

arXiv2024

Entropy solutions in $BV^s$ for a class of triangular systems involving a transport equation

In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in $L^\infty$ under optimal fractional $BV$ regularity of the initial data. Our results show that the critical fractional regularity is $s=1/3$. We also construct an initial data that is not in $BV^{1/3}$ but for which a blow-up in $L^\infty$ occurs, proving the optimality of our results.

math.AP↗

arXiv2024

Optimal regularity for all time for entropy solutions of conservation laws in $BV^s$

This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional $BV$ spaces. We show that optimality of the regularizing effect for the initial value problem from $L^\infty$ to fractional Sobolev space and fractional $BV$ spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover , we prove sharp smoothing in $BV^s$ for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

math.AP↗

arXiv2024

Global weak solutions of a Hamiltonian regularised Burgers equation

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term.

math.AP↗

arXiv2024

Hamiltonian regularisation of the unidimensional barotropic Euler equations

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh. This system is Galilean invariant, linearly non-dispersive and conserves formally an $H^1$-like energy. In this paper, we generalise this regularisation for the barotropic Euler system preserving the same properties. We prove the local (in time) well-posedness of the regularised barotropic Euler system and a periodic generalised two-component Hunterr-Saxton system. We also show for both systems that if singularities appear in finite time, they are necessary in the first derivatives.

math.AP↗

arXiv2024

Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

We prove in this note the local (in time) well-posedness of a broad class of $2 \times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond, Dutykh and Mitsotakis. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.

math.AP↗

arXiv2024

Regularizing effect for conservation laws with a Lipschitz convex flux

This paper studies the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, we generalize the well-known $\mathrm{BV}$ smoothing effect for $\mathrm{C}^2$ uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik, while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be dicontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized $\mathrm{BV}$ space, $\mathrm{BV}^Φ$.

math.AP↗

arXiv2020

Fractional $BV$ solutions for $2\times 2$ systems of conservation laws with a linearly degenerate field

The class of $2\times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional BV spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and $z$ corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates $ (w_0,z_0)$ are small in $ BV^s \times L^\infty $, $1/3 \leq s<1$. The restriction on the exponent $s$ is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant $z$ on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant $w$.

math.AP↗

arXiv2017

Kinetic formulation of a 2 $\times$ 2 hyperbolic system arising in gas chromatography

The PSA system commonly used in the context of gas-solid chromatography is reformulated as a single kinetic equation using an additional kinetic variable. A kinetic numerical scheme is built from this new formulation and its behavior is tested on solving the Riemann problem in different configurations leading to single or composite waves.

math.AP↗

arXiv2013

High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws

The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, (1994), about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a new definition of nonlinear flux is stated and compared to classical definitions. Then it is proved that the smoothness expected in Sobolev spaces cannot be exceeded.

math.AP↗

arXiv2013

Fractional BV spaces and first applications to scalar conservation laws

The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some "fractional $BV$ spaces" denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\R)$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\R)$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.

math.AP↗

arXiv2013

Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws

Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for entropy solutions of nonlinear scalar conservations laws . In this short paper we will restrict our attention to the simpler one-dimensional case. First, supercritical geometric optics lead to sequences of $C^\infty$ solutions uniformly bounded in the Sobolev space conjectured. Second we give continuous solutions which belong exactly to the suitable Sobolev space. In order to do so we give two new definitions of nonlinear flux and we introduce fractional $BV$ spaces.

math.AP↗

arXiv2012

Stability of a critical nonlinear neutral delay differential equation

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of the constant and periodic solutions. A complete analysis of the stability diagram is given. Lastly, existence of periodic solutions is discussed, involving a Diophantine condition on the period.

math.CA↗

arXiv2011

The Method of Strained Coordinates for Vibrations with Weak Unilateral Springs

We study some spring mass models for a structure having a unilateral spring of small rigidity $ε$. We obtain and justify an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: $T_\eps \sim \eps^{-1}$ as usual; or, for a new critical case, we can only expect: $T_\eps \sim \eps^{-1/2}$. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes" (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when $ε$ becomes larger.

math.DS↗

arXiv2010

Blow up at the hyperbolic boundary for a 2x2 system arising from chemical engineering

We consider an initial boundary value problem for a 2x2 system of conservation laws modeling heatless adsorption of a gaseous mixture with two species and instantaneous exchange kinetics, close to the system of Chromatography. In this model the velocity is not constant because the sorption effect is taken into account. Exchanging the roles of the x, t variables we obtain a strictly hyperbolic system with a zero eigenvalue. Our aim is to construct a solution with a velocity which blows up at the corresponding characteristic "hyperbolic boundary" {t = 0}.

math.AP↗

arXiv2009

Strong Stability with respect to weak limit for a Hyperbolic System arising from Gas Chromatography

We investigate a system related to a particular isothermal gas-solid chromatography process, called ?Pressure Swing Adsorption?, with two species and instantaneous exchange kinetics. This system presents the particularity to have a linearly degenerate eigenvalue: this allows the velocity of the gaseous mixture to propagate high frequency waves. In case of smooth concentrations with a general isotherm, we prove L1 stability for concentrations with respect to weak limits of the inlet boundary velocity. Using the Front Tracking Algorithm (FTA), we prove a similar result for concentrations with bounded variation (BV) under some convex assumptions on the isotherms. In both cases we show that high frequency oscillations with large amplitude of the inlet velocity can propagate without affecting the concentrations.

math.AP↗

arXiv2009

Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave

The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of the crack also increases with the amplitude of the forcing levels. The aim of the present theoretical study is to analyse these nonlinear processes under a larger range of nonlinear jump conditions. For this purpose, the problem is reduced to a nonlinear differential equation. The dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions. Bounds for the mean, maximum and minimum values of the solution are presented. Lastly, periodic forcing with a null mean value is addressed. In that case, a result about the mean dilatation of the crack is obtained.

physics.class-ph↗