---------------------------------------------------------------------- Abstracts of the lectures already available: ---------------------------------------------------------------------- TWO MODELS FROM NONLINEAR OPTICS Philip Holmes Program in Applied and Computational Mathematics and Department of Mechanical and Aerospace Engineering, Princeton University. I will discuss recent joint work with J. Nathan Kutz, Alex Mielke, and others in which variants and derivatives of the nonlinear Schroedinger equation (NLS), describing the propagation of light in optical fibers, are studied. In the first, we study a model of a broadband laser in the form of an NLS with non-local terms describing the averaged properties of a quantum mirror, the saturable Bragg reflector. We prove existence-uniqueness results and study the bifurcations and stability of certain "chirped soliton" solutions, comparing them with experimental results from W. Knox's group at Lucent Technologies. In the second, we derive a planar mapping approximating variations in amplitude and phase of a pulse propagating in a lossless optical fiber with periodically varying dispersion. The map's behavior agrees well with simulations of the periodically switched NLS due to S. Evangelides. We analyse the bifurcations of fixed points and global dynamics with a view to describing pulse modulation properties. --------------------------------------------------------------------- WELL-POSEDNESS OF THE NAVIER-STOKES EQUATIONS Hideo Kozono Mathematical Institute Tohoku University Sendai, 980-8578 JAPAN In this lecture, we will show first that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of strong solutions to the Navier-Stokes equations. Our result will be applied to the criterion on uniqueness and regularity of weak solutions in the marginal class of Serrin's. For the proof, we will establish various bilinear estimates in the BMO norm by means of the symbol calculus given by Coifman-Meyer. Then we will next deal with the similar blow-up phenomena to the Euler equations. To this end, we will introduce a critical Sobolev inequality of logarithmic type in the BMO and the Besov spaces which may be regarded as generalization of Brezis-Wainger's. --------------------------------------------------------------------- DIFFUSION AND CROSS-DIFFUSION IN PATTERN FORMATION Wei-Ming-Ni School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street S.E. Minneapolis, MN 554555 In this lecture series I intend to illustrate the role of diffusion and cross-diffusion in nonlinear systems from the point of viewof pattern formation. Using examples including Turing's "diffusion- driven instability" and the classical Lotka-Volterra competition model, I shall discuss the emergence of spatially inhomogeneous patterns as the diffusion rates and/or the cross- diffusion rates vary. Qualitative properties of these patterns will be included, and, the stability and dynamics will be emphasized. --------------------------------------------------------------------- A SURVEY OF THE KINETIC APPROACH TO CONSERVATION LAWS B. Perthame Ecole Normale Sup\'erieure, DMA 45, rue d'Ulm F 75230 Paris C\'edex 05 The kinetic approach to conservation laws concerns various aspects of the theory of hyperbolic systems. For those with a rich enough family of entropies, it allows to give a kinetic formulation which is a way to represent by a single equation the whole family of entropies. For scalar conservation laws it turns out to be a powerful tool to understand their mathematical structure and unify in a simple formalism most of the theory. As an example remains the proof of Sobolev regularizing effects for nondegenerate fluxes in multidimensional scalar conservation laws. Many open theoretic questions are still open which will be mentioned during the talk. More general, and useful for applications, is the derivation of stable numerical schemes for systems of gas dynamics type. As an example, we will explain how the kinetic schemes furnish a natural answer to the question of deriving numerical methods which preserv equilibriums for the Saint-Venant model of shallow water. ---------------------------------------------------------------------- GLOBAL EXISTENCE FOR SEMILINEAR PARABOLIC PROBLEMS Pavol Quittner \'UAM MFF UK Mlynsk\'a dolina SK-84 215 Bratislava, Slovakia In the first part, we derive sufficient conditions for global existence of solutions of semilinear parabolic problems in a smoothly bounded domain $\Omega\subset R^n$. These conditions require the boundedness of solutions in $L_p$-spaces. Our examples include some weakly coupled systems, problems involving measures, nonlinear boundary conditions and nonlocal nonlinearities. In the second part, we use the results mentioned above for the proof of a priori estimates of global solutions of some superlinear parabolic problems and we discuss some applications of these estimates. ----------------------------------------------------------------------