Asymptotic Analysis

In this paper we investigate the asymptotic behavior of the solutions of the one-dimensional generalized porous-thermo-elasticity problem. First, we prove that when only thermal damping is present the decay of solutions is slow (generically). Second, we show that if the porous dissipation is also present the decay of the solutions is controlled by an exponential.

In this paper we study the asymptotic behavior of the viscoelastic system with nondissipative kernels. We show that the uniform decay of the energy depends on the decay of the kernel, the positivity of the kernel in t=0 and some smallness condition. That is, if the kernel g∈C²( ${\mathbb{R}}$ ⁺) with g(0)>0, decays exponentially to zero then the solution decays exponentially to zero. On the other hand, if the kernel decays polynomially as t^−p then the corresponding solutions also decays polynomially to zero with the same rate of decay.

We provide conditions ensuring the existence of a solution to a noncoercive minimization problem. These conditions are also necessary. They enlarge the applicability of previous results due to Baiocchi, Buttazzo, Gastaldi and Tomarelli and their extensions by Auslender. A particular attention is given to the case of quasiconvex objective functions.

We consider the following stationary Keller–Segel system from chemotaxis $\[\Delta u-au+u^{p}=0,\quad u>0\ \hbox{in}\ {\varOmega},\qquad \frac{\curpartial u}{\curpartial \nu}=0\quad \hbox{on}\ \curpartial {\varOmega},\]$ where Ω⊂ $\mathbb{R}$ ² is a smooth and bounded domain. We show that given any two positive integers K,L, for p sufficiently large, there exists a solution concentrating in K interior points and L boundary points. The location of the blow-up points is related to the Green function. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.

Recently, Krylov, Barles, and Jakobsen developed the theory for estimating errors of monotone approximation schemes for the Bellman equation (a convex Isaacs equation). In this paper we consider an extension of this theory to a class of non-convex multidimensional Isaacs equations. This is the first result of this kind for non-convex multidimensional fully non-linear problems.

To get the error bound, a key intermediate step is to introduce a penalization approximation. We conclude by (i) providing new error bounds for penalization approximations extending earlier results by, e.g., Bensoussan and Lions, and (ii) obtaining error bounds for approximation schemes for the penalization equation using very precise a priori bounds and a slight generalization of the recent theory of Krylov, Barles, and Jakobsen.

We show that under a smallness assumption on the L^∞ norm of the initial data, there is a propagation of any Sobolev regularity greater than −1/2 for some weak solutions of the critical dissipative quasi-geostrophic equation. We also prove the existence of solutions in a space close to the homogeneous Sobolev space $\dot{H}$ ¹, theses solutions are given by a fixed point theorem.

In a previous work [N. Ben Abdallah and C. Negulescu, A one-dimensional transport model with small coherence lengths, Transp. Theory and Stat. Phys. 31(4–6) (2002), 559–578] a one-dimensional transport model accounting for both, quantum phenomena and smallness of coherence lengths, has been analyzed. In the limit of infinitely small coherence lengths, this model leads to the classical Vlasov equation. In this paper the two-dimensional situation is considered by adding to the one-dimensional transport direction a transversal confined one. The transport direction is splitted into several regions of a size comparable to the coherence length. A quantum model, based on the 2D Schrödinger equation, describes the electron transport within each cell, while on larger distances, statistics defined on the interfaces of the cells, determine the electron motion. Reflection and transmission coefficients, deduced from the wavefunctions, solutions of the Schrödinger equation, relate neighboring statistics with one another. In the limit of small coherence lengths, we obtain a collisionless subband model, which is classical in the transport direction and keeps quantum features in the confined one.

We study how three-dimensional linearized elasticity for thin plates can be approximated by a two-dimensional projection. The classical approach using formal asymptotic expansions in powers of the thickness in the Hellinger–Reissner formulation, only provides error estimates in the H¹ norm for the displacements, assuming at least L² regularity for the applied forces, plus additional regularity for some components. Here we make use of elliptic regularity theory. We prove a 3d–2d interior error estimate between the 3d displacement solution and its 2d projection. Moreover the constants involved in our estimate are independent on the particular geometry of the plate. Our approach yields an H² error estimate, assuming only L² regularity for the applied forces, which is optimal from the point of view of elliptic regularity theory. We also obtain interior W^k,p and C^k,α error estimates.