De Bouard Research Articles

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Abstract We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

White noise driven Ostrovsky equation

The current paper is devoted to the Cauchy problem for the white noise driven Ostrovsky equation with positive dispersion. We obtain the local well-posedness for the initial data u0(⋅,ω)∈Hs(R) (a.e. ω∈Ω) which is F0-measurable with s>−34 and Φ∈L20,s∩L20(L2(R),H˙s,−12+ϵ(R)), and the globally well-posedness for the initial data u0(⋅,ω)∈L2(R) (a.e. ω∈Ω) which is F0-measurable and Φ∈L20,0∩L20(L2(R),H˙0,−12+ϵ(R)). The key ingredients that we used in this paper are some bilinear estimates in Xs,b-type spaces given in subsection 2.1, Itô formula, the BDG inequality and stopping time techniques as well as frequency truncated technique. Our method can be applied to the case γ=0, which is just the KdV equation, thus, our result improves the local well-posedness result of A. de Bouard, A. Debussche and Y. Tsutsumi (1999) [3].

Generalized KdV equation subject to a stochastic perturbation

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.

Stochastic nonlinear Schrödinger equations

This paper is devoted to the well-posedness of stochastic nonlinear Schrödinger equations in the energy space H1(Rd), and is a continuation of our recent work (Barbu et al., 2014). We allow general complex coefficients in the noise, thus covering both the conservative and non-conservative case, the latter being important to include quantum measurement effects in the underlying physical model. We consider both focusing and defocusing nonlinearities and prove global well-posedness in H1(Rd), including also the pathwise continuous dependence on initial conditions, with exponents of the nonlinearity in exactly the same (sharp) range as in the deterministic case. In particular, in the special conservative case, this work improves earlier results in de Bouard and Debussche (2003), where the critical exponents could not be reached for d≥6 in the defocusing and for d≥4 in the focusing case respectively. Moreover, the local existence, uniqueness and blowup alternative are also established for the energy-critical case. The approach presented here is mainly based on the rescaling approach already used in Barbu et al. (2014) to study the L2 case and also on the Strichartz estimates established in Marzuola et al. (2008) for large perturbations of the Laplacian.

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

On the Stochastic Strichartz Estimates and the Stochastic Nonlinear Schrödinger Equation on a Compact Riemannian Manifold

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

The cult of images in light of pictorial graffiti at Doué-la-Fontaine

In the late 1960s the archeologist Michel de Bouard excavated a motte at Doue-la-Fontaine, near Saumur. Buried within the earthen mound was a stone edifice that, between the mid-tenth and early eleventh centuries, had been transformed from a single-storey aristocratic residence to a multi-storey tower better suited to new military needs. De Bouard there discovered pictorial graffiti, incised in rough plaster across a wall in the blinded ground storey; among them were several unusually elaborate compositions. Building on his perspicacious analysis of the physical evidence, I situate the graffiti, dating from c.1000, in the wider matrix of contemporary visual and religious culture. The material enriches our understanding of a major historical phenomenon for which the millennial era is a watershed, namely the emergence and proliferation of cult images.

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in A. de Bouard, A. Debussche (2010) [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.

Asymptotics of surface waves over random bathymetry

This paper addresses the propagation of free surface water waves over a variable seabed in the long wavelength scaling regime. We consider the situation in which the bathymetry is given by a stationary random process which has a correlation length substantially shorter than the wavelength of the principal surface wave components. An asymptotic description shows that the water waves problem is modeled by an effective system of equations that is related to the KdV, however in a reference frame given in terms of random characteristic coordinates, and in addition with a random amplitude modulation and random scattered component. The resulting random processes are strongly correlated and have canonical limits due to the Donsker invariance principle. Our analysis is based on the Hamiltonian description of water waves and long wave perturbation theory and a new criterion for asymptotic expansions of partial differential equations with rapidly varying coefficients. In this paper we give a detailed analysis of the transformation to random characteristic coordinates and the asymptotic form of the resulting transformed partial differential equations. A companion paper (de Bouard, A., Craig, W., Diaz-Espinosa, O., Guyenne, P., Sulem, C., Long wave expansions for water waves over random topography, Nonlinearity 21 (2008), 2143-2178) analyses in detail the asymptotic behavior of the resulting expression for solutions, and their consistency with the derivation of the effective model equations.

Invariant measures for a stochastic nonlinear Schroedinger equation

We will prove the existence of an invariant measure for a nonlinear Schrodinger equation with random noise in R n . The existence of solutions of the Cauchy problem in H 1 - and L 2 -settings was established by de Bouard and Debussche [4, 5]. Here we discuss only the defocusing equation with a zero-order dissipation. The proof for the existence of an invariant measure is based on various energy estimates and the approximation scheme to construct solutions, which are also crucial for the existence of global solutions to the Cauchy problem.

Non existence of L2?compact solutions of the Kadomtsev?Petviashvili II equation

We prove that there is no nontrivial solution of the Kadomtsev–Petviashvili II equation (KP II equation) \({{ (u_t+u_{{xxx}}+uu_x)_x+u_{{yy}}=0,\quad (x,y)\in {{\bf{ R}}}^2, }}\) which is L 2 compact (i.e. uniformly localized in L 2 norm) and travel to the right in the x variable. This result extends the previous work of de Bouard and Saut [3] stating that there is no traveling wave solution for the KP II equation. The proof uses a monotonicity property of the L 2 mass for solutions of the KP II equation (similar to the one for the KdV equation [12], [14]) and two virial type relations. The result still holds for some natural generalizations of the KP II equation (general nonlinearity, higher dispersion) and does not rely on the integrability of the equation.

The Archaeology of Monasticism: A Survey of Recent Work in France, 1970-1987

Recognition of medieval archaeology as a distinct field, worthy of study in its own right, began in France in the 1950s when Michel de Bouard established the Centre de Recherches Archeologiques Medievales (CRAM) at the Universite de Caen.' Development of the field accelerated in the 1960s with the establishment of the Laboratoire d'Archeologie Medievale under the direction of Gabrielle Demians d'Archimbaud at the Universite de ProvenceAix and with the creation of formal academic programs at Caen, Aix, and several other universities. It is important to note that the development of medieval archaeology in France occurred regionally and that research and study programs were initiated with a strong regional focus. For example, Aix concentrated on deserted villages, early monasteries, and ceramics, while Caen focused on Frankish cemeteries and medieval castles. The appearance in 1971 of the annual bulletin Archeologie m'dievale marked a watershed for the field. Archeologie m'dievale, first published by CRAM, began in 1973 to receive subvention from the Centre National de Recherches

Does the dislocation velocity in germanium depend on optical illumination?

The effect of optical illumination on the dislocation velocity in germanium was studied at 370°C and 450°C using a carbon arc lamp. No effect was found in contradiction to the results of Fortini and de Bouard.