On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

We observe that the comparison result of Barles–Biton–Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow. To cite this article: J. Chen, C. Pang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A numerical study of the null boundary controllability of a convection diffusion equation

In this paper we study numerically the cost of the null controllability of a linear control parabolic 1-D equation as the diffusion coefficient tends to 0. For this linear control parabolic 1-D equation, we know from a prior work by J.-M. Coron and S. Guerrero (2005), that, when the diffusion coefficient tends to 0, for a small controllability time, the norm of the optimal control tends to infinity and that, if the controllability time is large enough, this norm tends to 0. For controllability times which are not covered by this work, we estimate numerically the norm of the optimal control as the diffusion coefficient tends to 0. To cite this article: A. Salem, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Zero-noise solutions of linear transport equations without uniqueness: an example

We consider a classical one-dimensional example of linear transport equation without uniqueness of weak solutions. Under a suitable multiplicative noise perturbation, the equation is well posed. We identify the two solutions of the deterministic equation obtained in the zero-noise limit. In addition, we prove that the zero-viscosity solution exists and is different from them. To cite this article: S. Attanasio, F. Flandoli, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the resonance problem with nonlinearity which has arbitrary linear growth

This paper deals with the nonlinear two-point boundary value problem at resonance. Even nonlinearities g with an arbitrary linear growth in +∞ (resp. −∞) may be considered but only on the cost of the corresponding bound on their linear growth at −∞ (resp. +∞). It generalizes the previous results in this direction obtained by M. Schechter, J. Shapiro, and M. Snow (Trans. Amer. Math. Soc. 241 (1978), 69–78), L. Cesari and R. Kannan (Proc. Amer. Math. Soc. 88 (1983), 605–613), and S. Ahmad (Proc. Amer. Math. Soc. 93 (1984), 381–384).     

Contrôlabilité à zéro avec contraintes sur le contrôle

We study a problem of null-controllability for the parabolic heat equation with linear constraints on the control. The main tool used to solve the problem of existence and convergence is an observability inequality of Carleman type, which is ‘adapted’ to the constraints. We then apply the obtained results to the sentinels theory of Lions. To cite this article: O. Nakoulima, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Convergence of linear finite elements for diffusion equations with measure data

We show here the convergence of the linear finite element approximate solutions of a diffusion equation to a weak solution, with weak regularity assumptions on the data. To cite this article: T. Gallouët, R. Herbin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sobolev weak solutions for parabolic PDEs and FBSDEs

This Note is devoted to the representation of Sobolev weak solutions to quasi-linear parabolic PDEs with monotone coefficients via FBSDEs. One distinctive character of this result is that the forward component of the FBSDE is coupled with the backward variable. To cite this article: F. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Parallelization in time through tensor-product space–time solvers

In this Note, we extend the fast tensor-product algorithm for the simulation of time-dependent partial differential equations. We use the natural tensorization of the space–time domain to propose, after discretization, a set of independent problems, each one with the complexity of a single steady problem. This allows for an efficient parallel implementation that is already interesting on small architectures, but that can also be combined with standard domain-decomposition-based algorithms providing a further direction of parallelism on large computer platforms. Preliminary numerical simulations are presented for a one-dimensional unsteady heat equation. To cite this article: Y. Maday, E.M. Rønquist, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Application of the exact null controllability of the heat equation to moving sets

We study the lagrangian controllability of the heat equation in several dimensions. In dimension one, we prove that any pairs of intervals are diffeomorphic through the flow of the solution of the heat equation via an adequate control. In higher dimensions we prove a similar controllability result for the flow of the gradient of the solution in a radial case in arbitrary finite time, and for convex domains in a sufficiently large time. To cite this article: T. Horsin Molinaro, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Hypocoercivity for kinetic equations with linear relaxation terms

This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion. To cite this article: J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Nulle contrôlabilité régionale d'une équation de type Crocco linéariséeRegional null controllability of a linearized Crocco type equation

We are interested in controllability problems of equations coming from a boundary layer model. This problem is described by a degenerate parabolic equation (a linearized Crocco type equation) where phenomena of diffusion and transport are coupled.First we give a geometric characterization of the influence domain of a locally distributed control. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space–time domain and Carleman type estimates along characteristics. To cite this article: P. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 581–584.     

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where ?: C([a, b];?) → L([a, b];?) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.     

Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics

We present here an extension to any order of accuracy of the schemes proposed in Daru and Tenaud [J. Comput. Phys. 193 (2) (2004) 563–594] for the linear advection equation in 1D. Such schemes are then used for a high-order generalization of the Godunov method in the case of the wave equation and the locally linearized Euler equations. To cite this article: S. Del Pino, H. Jourdren, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the decay rate of solutions of linear parabolic equations for infinitely increasing time

The first boundary value problem with zero boundary values is considered for a one-dimensional linear parabolic equation. If the equation is sufficiently close to the heat equation, the rate of decreasing for the solution is connected with the number of zero level lines of the solution, nonvanishing for all values of time. Bibliography: 4 titles. Translated from Trudy Seminara imeni I.G. Petrovskogo, No. 17, pp. 118–127, 1994.     

Les équations de la chaleur et de Poisson pour le laplacien généralisé de Jacobi–Dunkl

We show that the heat equation for the Jacobi–Dunkl operator, has a solution in terms of a semigroup of Markovian operators with strictly positive kernel. This result is used to solve the Poisson equation and to introduce a new class of Markov processes on the real line. To cite this article: F. Chouchene et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Travelling Wavefronts in Reaction‐Diffusion Equations with Convection Effects and Non‐Regular Terms

This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction‐diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c ≥ c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a non‐linear convection effect. Moreover, we do not require the main non‐linearity g to be a regular C¹ function; in particular we are able to treat both the case when g′(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g′(0) = +∞. Our results generalize previous ones due to Aronson and Weinberger [Adv. Math. 30 (1978), pp. 33–76 ], Gibbs and Murray (see Murray [Mathematical Biology, Springer‐Verlag, Berlin, 1993 ]) and McCabe , Leach and Needham [SIAM J. Appl. Math. 59 (1998), pp. 870–899 ]. Finally, we obtain our conclusions by means of a comparison‐type technique which was introduced and developed in this framework in a recent paper by the same authors.     

Properties of the Fast Diffusion Equation and Its Multidimensional Exact Solutions

We prove invariance of the fast diffusion equation in the two-dimensional coordinate space and give its reduction to a one-dimensional analog in the space variable. Using these results, we construct new exact multidimensional solutions which depend on arbitrary harmonic functions. As a consequence, we obtain new exact solutions to the well-known Liouville equation, the stationary analog of the fast diffusion equation with a linear source. We consider some generalizations to the case of systems of quasilinear parabolic equations.     

Dynamics of multiple degree Ginzburg–Landau vortices

For the two-dimensional complex parabolic Ginzburg–Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff–Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).