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Benjamin Texier

Publications and source records attributed to Benjamin Texier.

At least 19 recordsLinked to original sources
arXiv2021

On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part two

We continue our study of initial-value problems for fully nonlinear systems exhibiting strong or weak defects of hyperbolicity. We prove that, regardless of the initial Sobolev regularity, the initial-value problem has no local $H^s$ solutions, for $s > s_0 + d/2,$ if the principal symbol has a strong, or even weak, defect of hyperbolicity, and the purely imaginary eigenvalues of the principal symbol are semi-simple and have constant multiplicity. The index $s_0 > 0$ depends on the severity of the defect of hyperbolicity. These results recover and extend previous work from G. M\'etivier [{\it Remarks on the well posedness of the nonlinear Cauchy problem,} 2005], N.Lerner, Y. Morimoto, C.-J. Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, 2010] and N. Lerner, T. Nguyen, B. Texier, {\it The onset of instability in first-order systems}, 2018]

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arXiv2021

Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system

We show that space-time resonances induce high-frequency Raman instabilities in the two-fluid Euler-Maxwell system describing laser-plasma interactions. A consequence is that the Zakharov WKB approximation to Euler-Maxwell is unstable for non-zero group velocities. A key step in the proof is the reformulation of the set of resonant frequencies as the locus of weak hyperbolicity for linearized equations around the WKB solution. We analyze those linearized equations with the symbolic flow method. Due to large transverse variations in the WKB profile, the equation satisfied by the symbolic flow around resonant frequencies is a linear {\it partial} differential equation. At space-time resonances corresponding to Raman instabilities, we observe a fast growth of the symbolic flow, which translates into an instability result for the original system. The strongest instability is caused by backscattered Raman waves.

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arXiv2021

Dispersive regularization for phase transitions

We introduce a dispersive regularization of the compressible Euler equations in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive regularization is of Schroedinger type. In particular, the regularized system is complex-valued. It has a conservation law, which, for real unknowns, is identical to the energy of the unregularized physical system. The regularized system supports high-frequency solutions, with an existence time or an amplitude which depend strongly on the pressure law.

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arXiv2020

On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part one

We prove that, for first-order, fully nonlinear systems of partial differential equations, under an hypothesis of ellipticity for the principal symbol, the Cauchy problem has no solution within a range of Sobolev indices depending on the regularity of the initial datum. This gives a new and greatly detailed proof of a result of G. M\'etivier [{\it Remarks on the Cauchy problem}, 2005]. We then extend this result to systems experiencing a transition from hyperbolicity and ellipticity, in the spirit of recent work by N. Lerner, Y. Morimoto, and C.-J. Xu, [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, 2010], and N. Lerner, T. Nguyen and B. Texier [{\it The onset of instability in first-order systems}, 2018].

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arXiv2016

On nonlinear stabilization of linearly unstable maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For G\^ateaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fr\'echet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fr\'echet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.

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arXiv2015

A stability criterion for high-frequency oscillations

We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition. The analysis relies, in the unstable case, on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.

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arXiv2015

Time-delayed instabilities in complex Burgers equations

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, Amer.~J.~Math.~2010] proved that only analytical data generate local $C^2$ solutions. The corresponding instabilities are however not observed numerically; rather, numerical simulations show an exponential growth only after a delay in time. We argue that numerical diffusion is responsible for this time delay, as we prove that for Burgers equations in the torus with small viscosity and a complex forcing, oscillating data generate solutions which grow linearly in time before growing exponentially. Numerical simulations illustrate the results.

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arXiv2015

The onset of instability in first-order systems

We study the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linear Lax-Mizohata theorem, was explained by G. M\'etivier in [{\it Remarks on the well-posedness of the nonlinear Cauchy problem}, Contemp.~Math.~2005]. In the present paper, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under such an hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto and C.-J. Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, American J.~Math.~2010] on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein-Gordon-Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, introduced by B.~Texier in [{\it Approximations of pseudo-differential flows}, Indiana Univ. Math. J.~2016].

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arXiv2014

Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations

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arXiv2014

Approximations of pseudo-differential flows

Given a classical symbol $M$ of order zero, and associated semiclassical operators ${\rm op}_\varepsilon(M),$ we prove that the flow of ${\rm op}_\varepsilon(M)$ is well approximated, in time $O(|\ln \varepsilon|),$ by a pseudo-differential operator, the symbol of which is the flow $\exp(t M)$ of the symbol $M.$ A similar result holds for non-autonomous equations, associated with time-dependent families of symbols $M(t).$ This result was already used, by the author and co-authors, to give a stability criterion for high-frequency WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here two further applications: sharp semigroup bounds, implying nonlinear instability under the assumption of spectral instability at the symbolic level, and a new proof of sharp G\r{a}rding inequalities.

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arXiv2013

From Newton to Boltzmann: hard spheres and short-range potentials

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the characteristic length of interaction $\e$ simultaneously goes to $0,$ in the Boltzmann-Grad scaling $N \e^{d-1} \equiv 1.$ The time of validity of the convergence is a fraction of the average time of first collision, due to a limitation of the time on which one can prove uniform estimates for the BBGKY and Boltzmann hierarchies. Our proof relies on the fundamental ideas of Lanford, and the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina. The main novelty here is the detailed study of pathological trajectories involving recollisions, which proves the term-by-term convergence for the correlation series expansion.

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arXiv2011

Nash--Moser iteration and singular perturbations

We present a simple and easy-to-use Nash--Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter $\eps\to 0.$ The novel feature is to allow loss of powers of $\eps$ as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems.

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arXiv2010

Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions

We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.

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arXiv2009

Existence of quasilinear relaxation shock profiles

We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude quasilinear relaxation shocks in the general case that the profile ODE may become degenerate. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman--Enskog expansion and the macro--micro decomposition of Liu and Yu. In the quasilinear case, however, we find it necessary to apply a parameter-dependent Nash-Moser iteration to close the analysis, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.

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arXiv2007

Galloping instability of viscous shock waves

Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincaré--Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.

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arXiv2006

Hopf bifurcation of viscous shock wave solutions of compressible Navier-Stokes equations and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier--Stokes (cNS) and magnetohydrodynamics (MHD) equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in $H^s$ for data in $H^s$. We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.

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arXiv2006

Pointwise Green function bounds and stability of combustion waves

Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier--Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, % under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu--Ying and Tesei--Tan for Majda's model and the reactive Navier--Stokes equations, respectively.

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