Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Existence and nonuniqueness of solutions to a functional-differential equation

We examine the functional-differential equation Δu(x) — div(u(H(x))f (x)) = 0 on a torus which is a generalization of the stationary Fokker-Planck equation. Under sufficiently general assumptions on the vector field f and the map H, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.     

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore

We consider an ergodic Hamilton–Jacobi–Bellman equation coming from a stochastic control problem in which there are exactly k points where the dynamics vanishes and the Lagrangian is minimal. Under a stabilizability assumption, we state that the solutions of the ergodic equation are uniquely determined by their value on these k points, and that the set of solutions is sup-norm isometric to a non-empty closed convex set whose dimension is less or equal to k. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

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).     

Self-similar solutions with fat tails for a coagulation equation with nonlocal drift

We investigate the existence of self-similar solutions for a coagulation equation with nonlocal drift. In addition to explicitly given exponentially decaying solutions we establish the existence of self-similar profiles with algebraic decay. To cite this article: M. Herrmann et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Representation theorems for generators of backward stochastic differential equations

It is proved that the generator g of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs if and only if g is a Lebesgue generator. To cite this article: L. Jiang, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On splitting the area under curves II

This paper continues the author's work [3, S. Minsker, J. Differential Equations, 26, No. 3 (1977), 443–457.] on an area-splitting problem leading to the functional differential equation $\text{a′(a(x)) =}\text{a(x)}\text{x}$. This equation is dealt with by transforming it into the linear equation ψ′(x) = ψ(x + c), for which positive solutions on (?∞, ?c) are sought.     

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).     

Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

Symmetry reductions and exact solutions of shallow water wave equations

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,$$ whereα andβ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by settingu _x =U, have been discussed in the literature. The caseα=2β was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math.,53 (1974), 249), who showed that this case was solvable by inverse scattering through a second-order linear problem. This case and the caseα=β were studied by Hirota and Satsuma (J. Phys. Soc. Japan,40 (1976), 611) using Hirota's bi-linear technique. Further, the caseα=β is solvable by inverse scattering through a third-order linear problem. In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole (J. Math. Mech,18 (1969), 1025). The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) withα=β which possess a rich variety of qualitative behaviours. These solutions all like a two-soliton solution fort < 0 but differ radically fort > 0 and may be viewed as a nonlinear superposition of two solitons, one travelling to the left with arbitrary speed and the other to the right with equal and opposite speed. These families of solutions have important implications with regard to the numerical analysis of SWW and suggests that solving (1) numerically could pose some fundamental difficulties. In particular, one would not be able to distinguish the solutions in an initial-value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours. We compare the two-soliton solutions obtained using the nonclassical method to those obtained using the singular manifold method and Hirota's bi-linear method. Further, we show that there is an analogous nonlinear superposition of solutions for two (2+1)dimensional generalisations of the SWW Equation (1) withα=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.     

Essential unitarization of symplectics and applications to field quantization

Symplectic operators satisfying generic and group-invariant (spectral) positivity conditions are studied; the theory developed is applied and illustrated to determine the unique invariant frequency decomposition (equivalently, linear quantization with invariant vacuum state) of the Klein-Gordon equation in non-static spacetimes. Let (H, Ω) be any linear topological symplectic space such that there exists a real-linear and topological isomorphism of H with some complex Hilbert space carrying Ω into the imaginary part of the scalar product. Then any bounded invertible symplectic S ∈ Sp(H) (resp. bounded infinitesimally symplectic A ∈ sp(H)) which satisfies $\text{Ω(Sv, v) > 0}$ (resp. $\text{Ω(Av, v) > 0}$) for all nonzero v ω H, where S + I is invertible, is realized uniquely and constructively as a unitary (resp. skewadjoint) operator in a complex Hilbert space which depends in general on the operator and typically only densely intersects H. The essentially unique weakly and uniformly closed invariant convex cones in sp(H) are determined, extending previously known results in the finite-dimensional case. A notion of “skew-adjoint extension” of a closed semi-bounded infinitesimally symplectic operator is defined, strictly including the usual notion of positive self-adjoint extension in a complex Hilbert space; all such skew-adjoint extensions are parametrized, as in the von Neumann or Birman-Krein-Vishik theories. Finally, the unique complex Hilbertian structure—formulated on the space of solutions of the covariant Klein-Gordon equation in generic conformal perturbations of flat space—is uniquely determined by invariance under the scattering operator. The invariant Hilbert structure is explicitly calculated to first order for an infinite-dimensional class of purely time-dependent metric perturbations, and higher-order contributions are rigorously estimated.     

Anti symmetric solutions of non-linear laminar flow between parallel permeable disks

The equations describing similarity solutions for flow between infinite parallel permeable disks with equal rates of suction or injection at the walls is derived using the stream function. This leads to a fourth order non-linear Ordinary Differential Equation. This equation is shown to admit anti-symmetric solutions using the moving plane method. To cite this article: Adimurthi, A. Karthik, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

Étude d'un système non linéaire de Boussinesq–Stefan

We give a few existence results for solutions for a class of Boussinesq–Stefan systems, with suitable conditions on the forcing terms in the right-hand side of the momentum equation depending on the temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Existence de solutions pour une classe de systèmes non linéaires de Boussinesq

We give a few existence results of solutions for a class of Boussinesq systems, with suitable conditions on the right-hand side of the momentum equation, the forcing term depending on temperature. To cite this article: A. Attaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Propriétés probabilistes des processus GARCH périodiques

This Note examines the probabilistic structure of a GARCH-type stochastic difference equation with periodically time-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions, geometrically ergodic (in the periodic sense) and having finite higher-order moments. To cite this article: A. Bibi, A. Aknouche, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Résultats d'existence pour les écoulements réguliers de fluides viscoélastiques incompressibles à loi différentielle de type White–Metzner en dimension 3

This paper is concerned with incompressible viscoelastic fluids which obey a differential constitutive law of White–Metzner type. We establish the existence and uniqueness of local solutions in 3-D as well as the global existence of small solutions. We then deduce the existence and asymptotic stability of small periodic and stationary solutions. Finally, we prove that the 2-D results obtained in Hakim (J. Math. Anal. Appl. 185 (1994) 675–705) remain true without any restriction on the smallness of the retardation parameter which is the linking coefficient between the equation of velocity (Navier–Stokes equation) and the transport equation verified by the extra-stress tensor. To cite this article: L. Molinet, R. Talhouk, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

In the present contribution, we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity field u is ruled by the Navier–Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director field d is described by a variation of a parabolic Ginzburg–Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.