Session B: Fluid mechanic and coastal dynamic

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**A. C. Alvarez**

(Habana, Cuba)

(Habana, Cuba)

Solving Mild Slope equation in Matlab with

Domain decomposition method

(joint with M. Sarkis and D. Marchesin)

In this paper the mild slope equation is solved on Matlab environmental. The method take into account physical processes waves as diffraction, refraction reflection, shoaling, port resonance and breaking. Further the effects of wave dissipation by friction, nonlinear amplitude dispersion are include. The algorithm is based on minimization of nonlinear functional using damped Newton iteration with Armijo-Goldstein line search strategy. Each linear problem is solved with Finite Element Method. Domain decomposition method with Schur complement is used in the linear case.**__________________________________________________________**

**Tadayoshi Kano**

**(Kyoto, Japan)**

Tunamis on a deep open sea and on a gentle sloping beach

Tunamis as shallow water waves:

(1) The shallow water wave equations show that tunamis propagate on a deep open sea by, at least, the sound speed. (2) On a gentle sloping beach, the shallow water wave equations show that the diminution of propagation speed caused by the diminution of the water-depth results the deformation and, eventually, the break down of tunamis. The shallow water wave equation have a corresponding conservation law on a sloping beach and shock formation corresponding tunamis breaking on the shore.

(3) A mathematical justification for tunamis as shallow water waves is given.

(1) The shallow water wave equations show that tunamis propagate on a deep open sea by, at least, the sound speed. (2) On a gentle sloping beach, the shallow water wave equations show that the diminution of propagation speed caused by the diminution of the water-depth results the deformation and, eventually, the break down of tunamis. The shallow water wave equation have a corresponding conservation law on a sloping beach and shock formation corresponding tunamis breaking on the shore.

(3) A mathematical justification for tunamis as shallow water waves is given.

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**David Lannes**

**(Paris, France)**

A stability condition for two-fluid interfaces

The aim of this talk is to derive a nonlinear stability condition for the interface between two fluids. This stability condition generalized the nonlinear Taylor criterion for the one fluid case and the linear Kelvin condition for the two fluid case. We show that it can explain the persistance of various natural phenomena such as waves and internal waves.**__________________________________________________________**

**Andre Nachbin**

**(Rio de Janeiro, Brasil)**

Asymptotic issues regarding nonlinear wave/topography interaction

The asymptotic issues of interest regard both the PDEs' (i.e. reduced models) as well as the solution's point of view. In particular for problems involving the interaction of coastal waves with quite arbitrary topography profiles. Phenomena studied are the apparent diffusion and waveform inversion of long waves in the presence of highly disordered topographies. I will report on recent results concerning the formulation of accurate reduced (PDE) models, where accuracy concerns asymptotic properties of solutions in respect to the full potential theory model.**__________________________________________________________**

**Cristina Turner**

**(Cordoba, Argentina)**

Nonlinear stability of two-layer flows

We study the dynamics of two–layer, stratiﬁed shallow water ﬂows. This is a model in which two scenarios for eventual mixing of stratiﬁed ﬂows (shear-instability and internal breaking waves) are, in principle, possible. We ﬁnd that unforced ﬂows cannot reach the threshold of shear-instability, at least without breaking ﬁrst. This is a fully nonlinear stability result for a model of stratiﬁed, sheared ﬂow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable –smoothly– from hyperbolic initial conditions. If the characteristic ﬁelds depend smoothly on the system’s Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability. **__________________________________________________________**

**Baldomero Valiño Alonso**

**(La Habana, Cuba)**

Infinitely narrow soliton in shallow water equations

(joint with Amaury Álvarez Cruz, Instituto de Oceanología, Cuba and Alex Méril, Université des Antilles et Guyana, Guadeloupe, France)

(joint with Amaury Álvarez Cruz, Instituto de Oceanología, Cuba and Alex Méril, Université des Antilles et Guyana, Guadeloupe, France)

V.P.Maslov have shown that infinitely narrow solitons appear in Burgers’s equation as limit cases of singular solutions of Korteveg-De Fries’s equation. In this work we seek infinitely narrow solitons for shallow water equations, in the case when the profile of the depth has an isolated singular point. Following V. P. Maslov, we look for that singular solution as an asymptotical development with smooth coefficients in an appropriate algebra of generalized functions. Singular solutions of hyperbolic systems of partial differential equations in general position may have only a finite number of structuraly stable and self similar singular solutions. When the time evolves, the singular solution preserves the shape and structure of the initial condition. According to Maslov, the coeffcients of the formal asymptotic expansion of the singular solution satisfy an infinite chain of ordinary differential equations (the Hugoniot-Maslov chain corresponding to the singular solution involved). The truncation of the chain makes possible its aproximate calculation. From the point of view of numerical calculation, this approach is advantageous because it reduces the problem of solving numerically the nonlinear system of partial di¤erential equations to solving a closed system of ordinary differential equations. We interpret "infinitely narros solitons" as examples of simplified generalized functions in the sense of J. F. Colombeau and we use the rules of Colombeau’s differential algebras to do the calculations.

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Session C: Mean field games

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**Italo Capuzzo Dolcetta**

**(Roma, Italy)**

A numerical approach to mean field games

Mean ﬁelds games models introduced by Lasry-Lions describe through new systems of pde’s the asymptotic behavior of differential games in which the number of players tends to +∞. In the talk we discuss some recent work in collaboration with Y. Achdou and F. Camilli about numerical methods for the approximation of stationary and nonstationary versions of such models. Particular attention will be given to the optimal planning problem, in which the positions of a very large number of identical rational agents, with common value function, evolve from an initial given spatial density to a desired target density at the ﬁnal horizon time.