to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

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Nonlinear dispersive equations : local and global analysis

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrodinger equation iut + 1u = ±|u|2u for large spherically symmetric Lx(R) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=11&iss=6&rank=4

The cubic nonlinear Schrödinger equation in two dimensions with radial data

In this paper we study the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle.

Global solutions of nonlinear schrodinger equations

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schrodinger equations and derive a direct proof of Nakanishi's H 1 scattering result for every L 2 -supercritical nonlinearity. We also prove scattering below the energy space for a certain class of L 2 -supercritical equations. In this paper we obtain new 1 a priori estimates for solutions of the nonlinear Schrodinger equation in one and two dimension. We also provide a systematic way to obtain the known interaction a priori estimates for dimensions higher than three. These estimates are monotonicity formulae that take advantage of the conservation of the momentum of the equation. Due to the pioneering work (19), estimates of this type are referred to as Morawetz estimates in the literature. We then apply these estimates to study the global behavior of solutions to the nonlinear Schrodinger equation. To be more precise we want to study the global-in-time behavior of solutions to the following initial value problem

/pdf/tensor-products-and-correlation-estimates-with-applications-39rqpz1aix.pdf

Tensor products and correlation estimates with applications to nonlinear Schrödinger equations

We prove low regularity global well-posedness for the Id Zakharov system and the 3d Klein-Gordon-Schrodinger system, which are systems in two variables u : R d x x R t → C and n: R d x x R t → R. The Zakharov system is known to be locally well-posed in (u,n) ∈ L 2 x H -½ and the Klein-Gordon-Schrodinger system is known to be locally well-posed in (u, n) ∈ L 2 x L 2 . Here, we show that the Zakharov and Klein-Gordon-Schrodinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the L 2 norm of u and controlling the growth of n via the estimates in the local theory.

https://ams.org/journals/tran/2008-360-09/S0002-9947-08-04295-5/S0002-9947-08-04295-5.pdf

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H-s initial data, s > -1/2, and for any s(1) = 0. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data, then the solution of KdV (given by the L-2 theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the defocusing modified KdV equation on the torus for s > 1/2.

Global Smoothing for the Periodic KdV Evolution

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

In this paper we consider the Zakharov system with periodic boundary con- ditions in dimension one. In the rst part of the paper, it is shown that for xed initial data in a Sobolev space, the dierence of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and obtain analogous smoothing estimates. As a corollary we prove the existence and smoothness of global attractors in the energy space. u(x; 0) = u0(x)2 H s0 (T); n(x; 0) = n0(x)2 H s1 (T); nt(x; 0) = n1(x)2 H s1 1 (T);

/pdf/smoothing-and-global-attractors-for-the-zakharov-system-on-ehfv9xa2c3.pdf

Nikolaos Tzirakis

Papers

Tensor products and correlation estimates with applications to nonlinear Schrödinger equations

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

Global Smoothing for the Periodic KdV Evolution

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

Smoothing and global attractors for the Zakharov system on the torus