Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrodinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

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Endpoint Strichartz estimates

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Commutator estimates and the euler and navier‐stokes equations

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

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

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A bilinear estimate with applications to the KdV equation

This paper is concerned with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations. Also we shall study some applications of these smoothing properties to nonlinear problems and their link with restriction theorems for the Fourier transform and pointwise convergence results

Oscillatory integrals and regularity of dispersive equations

(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:

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Gustavo Ponce

Papers

Commutator estimates and the euler and navier‐stokes equations

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

A bilinear estimate with applications to the KdV equation

Oscillatory integrals and regularity of dispersive equations

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