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Strichartz estimates for water waves

Thomas Alazard, Nicolas Burq, Claude Zuily (2011)

Annales scientifiques de l'École Normale Supérieure

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ( η = 0 , ψ = 0 )).

Strichartz inequality for orthonormal functions

Rupert Frank, Mathieu Lewin, Elliott H. Lieb, Robert Seiringer (2014)

Journal of the European Mathematical Society

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

Structural Evolution of the Taylor Vortices

Tian Ma, Shouhong Wang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We classify in this article the structure and its transitions/evolution of the Taylor vortices with perturbations in one of the following categories: a) the Hamiltonian vector fields, b) the divergence-free vector fields, and c). the solutions of the Navier-Stokes equations on the two-dimensional torus. This is part of a project oriented toward to developing a geometric theory of incompressible fluid flows in the physical spaces.

Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law

Manuel Bernard, Stéphane Dellacherie, Gloria Faccanoni, Bérénice Grec, Yohan Penel (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce...

Study of a three component Cahn-Hilliard flow model

Franck Boyer, Céline Lapuerta (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we propose a new diffuse interface model for the study of three immiscible component incompressible viscous flows. The model is based on the Cahn-Hilliard free energy approach. The originality of our study lies in particular in the choice of the bulk free energy. We show that one must take care of this choice in order for the model to give physically relevant results. More precisely, we give conditions for the model to be well-posed and to satisfy algebraically and dynamically consistency...

Study of Anisotropic MHD system in Anisotropic Sobolev spaces

Jamel Ben Ameur, Ridha Selmi (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

Three-dimensional anisotropic magneto-hydrodynamical system is investigated in the whole space ℝ 3 . Existence and uniqueness results are proved in the anisotropic Sobolev space H 0 , s for s > 1 / 2 . Asymptotic behavior of the solution when the Rossby number goes to zero is studied. The proofs, where the incompressibility condition is crucial, use the energy method, an appropriate dyadic decomposition of the frequency space, product laws in anisotropic Sobolev spaces and Strichartz-type estimates.

Supercomplex structures, surface soliton equations, and quasiconformal mappings

Julian Ławrynowicz, Katarzyna Kędzia, Osamu Suzuki (1991)

Annales Polonici Mathematici

Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. The following results are obtained: (i) A field operator of Dirac type, which is called a Hurwitz operator, is introduced by use of a Hurwitz pair and its characterization is given (Theorem 1). (ii) A field equation of the elliptic Neveu-Schwarz model of superstring theory is obtained from the Hurwitz pair (⁴,³) (Theorem 2), and its counterpart connected with the Hurwitz triple ( 11 , 11 , 26 ) is mentioned....

Supersymmetry and Ghosts in Quantum Mechanics

Robert, Didier (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 81Q60, 35Q40.A standard supersymmetric quantum system is defined by a Hamiltonian [^H] = ½([^Q]*[^Q] +[^Q][^Q]*), where the super-charge [^Q] satisfies [^Q]2 = 0, [^Q] commutes with [^H]. So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative. On the other hand Pais-Ulhenbeck proposed in 1950 a model in quantum-field theory where the d'Alembert operator [¯] = [(∂2)/( ∂t2)] − Δx is replaced by fourth order operator [¯]([¯] + m2), in order to...

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