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11 results on '"Incompressible euler equations"'

1. From Newton's second law to Euler's equations of perfect fluids

Author

Mikaela Iacobelli, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), and ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)

Subjects
General Mathematics, FOS: Physical sciences, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Fluid dynamics, Coulomb, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Limit (mathematics), 0101 mathematics, Mathematical Physics, Physics, Heuristic, Applied Mathematics, 010102 general mathematics, Dynamics (mechanics), Mathematical Physics (math-ph), 010101 applied mathematics, Classical mechanics, Energy method, Euler's formula, symbols, Analysis of PDEs (math.AP)
Abstract

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

Published
2021
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2. The Limit $\alpha \to 0$ of the $\alpha$-Euler Equations in the Half-Plane with No-Slip Boundary Conditions and Vortex Sheet Initial Data

Author

Helena J. Nussenzveig Lopes, Adriana Valentina Busuioc, Milton D. Lopes Filho, Dragoş Iftimie, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010)

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Applied Mathematics, Mathematical analysis, Slip (materials science), Euler equations, Computational Mathematics, symbols.namesake, Incompressible flow, Vortex sheet, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Boundary value problem, Analysis, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions. We establish the existence of subsequences conv...

Published
2020
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3. Self-similar point vortices and confinement of vorticity

Author

Carlo Marchioro, Dragoş Iftimie, Équations aux dérivées partielles, analyse (EDPA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010)

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Toy model, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Vorticity, 01 natural sciences, Vortex, Physics::Fluid Dynamics, 010101 applied mathematics, Dimension (vector space), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Point (geometry), 0101 mathematics, Analysis, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the squ...

Published
2018
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4. Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

Author

Luca Nenna, Guillaume Carlier, Jean-David Benamou, Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Geodesic, FOS: Physical sciences, 010103 numerical & computational mathematics, Multi-marginal optimal transport, 01 natural sciences, Regularization (mathematics), symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Incompressible Euler equations, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Mathematics - Numerical Analysis, 0101 mathematics, MSC : 76M30, 65K10, Mathematics - Optimization and Control, Mathematical Physics, Mathematics, Entropic regularization, Applied Mathematics, Numerical analysis, Numerical Analysis (math.NA), Mathematical Physics (math-ph), Euler equations, 010101 applied mathematics, Sinkhorn algorithm, Computational Mathematics, Optimization and Control (math.OC), symbols, Compressibility, Preprint, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Generalized incompressible flows, Algorithm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 76M30, 65K10
Abstract

Starting from Brenier’s relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn’s algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem (see Arnaudon et al. in An entropic interpolation problem for incompressible viscid fluids, 2017, arXiv preprint arXiv:1704.02126 ). Numerical results in dimension one and two illustrate the feasibility of the method.

Published
2017

5. Existence of small loops in the Bifurcation diagram near the degenerate eigenvalues

Author

Coralie Renault, Taoufik Hmidi, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-13-BS01-0003,DYFICOLTI,DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces ( 2013 ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), ANR-13-BS01-0003,DYFICOLTI,DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces(2013), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
General Physics and Astronomy, Bifurcation diagram, 01 natural sciences, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, 0101 mathematics, Global structure, Relative equilibria, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Degenerate energy levels, Statistical and Nonlinear Physics, Euler equations, 010101 applied mathematics, symbols, Analysis of PDEs (math.AP)
Abstract

In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming a small loop provided that they are close enough. This confirms the numerical observations done in the recent work [10], 32 pages

Published
2016
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6. Analyticity and Gevrey-Class Regularity for the Second-Grade Fluid Equations

Author

Marius Paicu, Vlad Vicol, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics (USC), University of California [Los Angeles] (UCLA), University of California-University of California, Laboratoire de Mathématiques d'Orsay (LM-Orsay), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::Analysis of PDEs, FOS: Physical sciences, 01 natural sciences, Upper and lower bounds, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Gevrey class, Mathematical Physics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Fluid Dynamics (physics.flu-dyn), Physics - Fluid Dynamics, Radius, Condensed Matter Physics, 010101 applied mathematics, Computational Mathematics, Fluid equation, Persistence (discontinuity), Analysis of PDEs (math.AP)
Abstract

International audience; We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions that does not vanish as t → ∞, and which is independent of the Rivlin-Ericksen material parameter α. Applications to the damped incompressible Euler equations are also given.

Published
2010
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7. Stabilized Spectral Element Approximation of the Saint Venant System Using the Entropy Viscosity Technique

Author

Richard Pasquetti, Jean-Luc Guermond, Boyan Popov, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Control, Analysis and Simulations for TOkamak Research (CASTOR), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Department of Mathematics [Texas] (TAMU), Texas A&M University [College Station], Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
Saint venant, Spectral element method, Mathematical analysis, Euler system, 01 natural sciences, 6. Clean water, 010305 fluids & plasmas, Physics::Fluid Dynamics, 010101 applied mathematics, Entropy inequality, Classical mechanics, 13. Climate action, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, 0101 mathematics, Shallow water equations, ComputingMilieux_MISCELLANEOUS, Physics::Atmospheric and Oceanic Physics, Mathematics
Abstract

We consider the Saint Venant system (shallow water equations), i.e. an approximation of the incompressible Euler equations widely used to describe river flows, flooding phenomena or erosion problems. We focus on problems involving dry-wet transitions and propose a solution technique using the Spectral Element Method (SEM) stabilized with a variant of the Entropy Viscosity Method (EVM) that is adapted to treat dry zones.

Published
2015
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8. On the Global Existence for the Axisymmetric Euler-Boussinesq System in Critical Besov Spaces

Author

Samira Sulaiman, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
axisymmetric flows, General Mathematics, 010102 general mathematics, Mathematical analysis, Rotational symmetry, Mathematics::Analysis of PDEs, Vorticity, 01 natural sciences, global well-posedness, 010305 fluids & plasmas, Physics::Fluid Dynamics, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, critical Besov spaces, 0103 physical sciences, FOS: Mathematics, Euler's formula, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Uniqueness, 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ ${\rho}^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$ with $p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity., Comment: Asymptotic Analysis journal, (2011)

Published
2012
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9. On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework

Author

Danchin, Raphaël, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)

Subjects
Mathematics - Analysis of PDEs, 76B03, elliptic estimates, Incompressible Euler equations, transport equation, Besov spaces, FOS: Mathematics, nonlinear analysis, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], MSC 76B03, Fourier analysis, Analysis of PDEs (math.AP)
Abstract

The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case $B^{\frac Np+1}_{p,1}(\R^N).$ A continuation criterion in the spirit of the celebrated one by Beale-Kato-Majda for the classical Euler equations, is also proved. In contrast with the previous work dedicated to this system in the whole space, our approach is not restricted to the $L^2$ framework or to small perturbations of a constant density state: we just need the density to be bounded away from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients., 31 pages

Published
2009

10. Energy of tsunami waves generated by bottom motion

Author

Denys Dutykh, Frédéric Dias, Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques et de Leurs Applications (CMLA), École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Laboratoire de recheche conventionné MESO (LRC MESO), École normale supérieure - Cachan (ENS Cachan)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS), EU project TRANSFER, contract no. 037058, and Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])

Subjects
Tsunami wave, 010504 meteorology & atmospheric sciences, [SDU.STU.GP]Sciences of the Universe [physics]/Earth Sciences/Geophysics [physics.geo-ph], General Mathematics, General Physics and Astronomy, Motion (geometry), [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], Kinetic energy, 01 natural sciences, 010305 fluids & plasmas, shallow-water equations, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Physics - Geophysics, [PHYS.PHYS.PHYS-COMP-PH]Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph], Mathematics - Analysis of PDEs, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], 14. Life underwater, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Shallow water equations, 0105 earth and related environmental sciences, Physics, [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Nonlinear Sciences - Exactly Solvable and Integrable Systems, water waves, General Engineering, Mechanics, Energy budget, Nonlinear Sciences - Pattern Formation and Solitons, Physics - Atmospheric and Oceanic Physics, Free surface, Physics - Computational Physics, Energy (signal processing), tsunami energy
Abstract

In the vast literature on tsunami research, few articles have been devoted to energy issues. A theoretical investigation on the energy of waves generated by bottom motion is performed here. We start with the full incompressible Euler equations in the presence of a free surface and derive both dispersive and non-dispersive shallow-water equations with an energy equation. It is shown that dispersive effects only appear at higher order in the energy budget. Then we solve the Cauchy-Poisson problem of tsunami generation for the linearized water wave equations. Exchanges between potential and kinetic energies are clearly revealed., Comment: 20 pages, 12 figures. Accepted to Proceedings of the Royal Society A. Other authors papers and supporting material can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Published
2009
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11. Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

Author

Alex Mahalov, Basil Nicolaenko, François Golse, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Tempe, Arizona], Arizona State University [Tempe] (ASU), and C. Bardos, A. Fursikov

Subjects
Physics, Semi-implicit Euler method, Euler number (physics), 010102 general mathematics, Mathematical analysis, Incompressible Euler Equations, 01 natural sciences, 010305 fluids & plasmas, Euler equations, Euler's laws of motion, Physics::Fluid Dynamics, symbols.namesake, Riemann problem, Classical mechanics, 0103 physical sciences, (MSC) 35Q35, 76B03, 76U05, symbols, Euler's formula, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Rigid Body Dynamics, Homoclinic orbit, 0101 mathematics, Euler's equations, Enstrophy Bursts, Rotating Fluids
Abstract

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy.

Published
2008

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