Gustavo Ponce

A bilinear estimate with applications to the KdV equation

u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on ‖u0‖Hs . Equation in (1.1) was derived by Korteweg and de Vries [21] as a model for long wave propagating in a channel. A large amount of work has been devoted to the existence problem for the IVP (1.1). For instance, (see [9], [10]), the inverse scattering method applies to this problem, and, under appropriate decay assumptions on the data, several existence results have been established, see [5],[6],[14],[28],[33]. Another approach, inherited from hyperbolic problems, relies on the energy estimates, and, in particular shows that (1.1) is locally well posed in H(R) for s > 3/2, (see [2],[3],[12],[29],[30],[31]). Using these results and conservation laws, global (in time) well posedness in H(R), s ≥ 2 was established, (see [3],[12],[30]). Also, global in time weak solutions in the energy space H(R) were constructed in [34]. In [13] and [22] a “local smoothing” effect for solutions of (1.1) was discovered. This, combined with the conservation laws, was used in [13] and [22] to construct global in time weak solutions with data in H(R), and even in L(R). In [16], we introduced oscillatory integral techniques, to establish local well posedness of (1.1) in H(R), s > 3/4, and hence, global (in time) well posedness in H(R), s ≥ 1. (In [16] we showed how to obtain the above mentioned result by Picard iteration in an appropriate function space.) In [4] J. Bourgain introduced new function spaces, adapted to the linear operator ∂t+∂ 3 x, for which there are good “bilinear” estimates for the nonlinear term ∂x(u /2). Using these spaces, Bourgain was able to establish local well posedness of (1.1) in H(R) = L(R), and hence, by a conservation

Well-posedness of the initial value problem for the Korteweg-de Vries equation

(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces Hs(R) . We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [ 1 5] that the IVP (1. 1) is locally (resp. globally) well-posed in Hs with s > 3/2 (resp. s > 2). Roughly speaking, global well-posedness in Hs depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely:

On the ill-posedness of some canonical dispersive equations

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

Global existence of small solutions to a class of nonlinear evolution equations

On etudie le comportement global des solutions dune classe dequations devolution non lineaires. On considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations dequations lineaires dissipatives

Quadratic forms for the 1-D semilinear Schrödinger equation

This paper is concerned with 1-D quadratic semilinear Schr6dinger equations. We study local well posedness in classical Sobolev space HS of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.

Local regularity of nonlinear wave equations in three space dimensions

This note examines the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for three-dimensional nonlinear wave equations of the form [partial derivative][sub t][sup 2]u [minus] [Delta]u = G(u, Du), (1), u(0) = f, [partial derivative][sub t]u(0) = g, (2), where Du = ([partial derivative][sub t]u, [del][sub x]u). For nonintegral s, this requires the use of an inequality. Thus it is crucial to gain control of the L[sup [ell][minus]1] norm in time of the maximum norm of the gradient of the solution in space. This is usually done by using the Sobolev imbedding theorem which leads to the restriction s > n/2+1 for the Sobolev exponent. The authors show that in three space dimensions (the case to which they restrict themselves throughout the paper), the lower bound for the Sobolev exponent can be reduced from 5/2 to s([ell]) [identical to] max[l brace]2, (5[ell] [minus] 7)/(2[ell] [minus] 2)[r brace] when the nonlinearity G in (1) grows no faster than order [ell] in Du. Thus, as [ell] [yields] [infinity] the classical result s > 5/2 is approached. They also show that this is sharp in the sense that the quantity [parallel]f[parallel][sub H[sup s([ell])]] + [parallel]g[parallel][sub H[supmorexa0» s([ell])[minus]1]] is not sufficient, in general, to control the local existence time of solutions (for [ell] [ge] 3). The authors return to the spherically symmetric case at the end of the paper. For general nonlinearities, such a result just fails. It is shown how to apply instead a space-time estimate for the free solution due to Marshall and Pecher, as an extension of the work of Strichartz. 9 refs.«xa0less

Nonlinear Small Data Scattering for the Generalized Korteweg-de Vries Equation

Abstract We study the longtime stability of small solutions to the IVP for the generalized Korteweg-de Vries equation. We obtain a lower bound for the degrees of nonlinearity of the perturbation which guarantees that the small solutions of the nonlinear problem behave asymptotically like the solutions of the associated linear problem. This behavior allows us to establish a nonlinear scattering result for small perturbations with these degrees. For a class of small data we improve the value of this lower bound. The new crucial ingrediants in our proofs are the L p -decay estimates of the half derivative of the Airy kernel.

Regularity of solutions to nonlinear dispersive equations

On considere le probleme de Cauchy pour les equations dispersives non lineaires de la forme ∂ t u+iP(D)u=F(u), x∈R n , −∞<t<∞, u(x,0)=u 0 (x)

On the Persistent Properties of Solutions to Semi-Linear Schrodinger Equation

We study persistent properties of solutions of the semi-linear Schrödinger equations in weighted spaces.

Global existence for the critical generalized KdV equation

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation, u t + u xxx + u 4 u x = 0, x, t E R. The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space H 1 (R), i.e. solutions corresponding to data u 0 ∈ H s (R), s > 3/4, with ∥u 0 ∥ L 2 < ∥Q∥ L 2, where Q is the solitary wave solution of the equation.