Asymptotic Analysis

We study the Cauchy problem for a damped wave equation. Our main result is the ( N + 1 ) th order approximation of solutions by the solution of the corresponding parabolic equation for arbitrary N ∈ N in the L p framework. This means that our results can be applied to nonlinear damped wave equations.

In this paper we consider a linear KdV equation with a transport term posed on a finite interval with the boundary conditions considered by Colin and Ghidaglia. The main results concern the behavior of the cost of null controllability with respect to the dispersion coefficient when the control acts on the left endpoint. In particular, for any final time we prove that this cost grows exponentially as the dispersion coefficient vanishes and the transport coefficient is negative.

In this paper, we are interested in the study of the solution to a generalization of the Cahn–Hilliard equation endowed with Neumann boundary conditions. This model has, in particular, applications in biology and in chemistry. We show that the solutions blow up in finite time or exist globally in time. We further prove that the relevant, from a biological and a chemical point of view, solutions converge to a constant as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

In this paper, temporal decay estimates for weak solutions to the three dimensional generalized Navier–Stokes equations are established firstly. With these estimates at disposal, algebraic time decay for higher order Sobolev norms of small initial data solutions are obtained. The decay rates are optimal in the sense that they coincide with ones of the corresponding generalized heat equation.

In this article we consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in n-dimensional compact Riemannian manifolds, for d = 2 , 3 . The system we consider consists of a regularized family of Navier–Stokes equations for the fluid velocity u coupled with a convective Allen–Cahn equation for the order (phase) parameter ϕ. We discretize these equations in time using the implicit Euler scheme and we prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

In this paper we prove an asymptotic estimate, up to the second-order included, on the behaviour of the one-dimensional Allen–Cahn’s action functionals, around a periodic function with bounded variation and taking values in { ± 1 } . The leading term of this estimate justifies and confirms, from a variational point of view, the results of Fusco–Hale [Dyn. Diff. Equation 1 (1989), 75–94] and Carr–Pego [Comm. Pure Appl. Math. 42 (1989), 523–576] on the exponentially slow motion of metastable patterns coexisting at the transition temperature.