scholarly journals Recent advances on the global regularity for irrotational water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170089 ◽  
Author(s):  
A. D. Ionescu ◽  
F. Pusateri
Keyword(s):  
Water Waves ◽  
Global Regularity ◽  
Local Existence ◽  
Three Dimensions ◽  
Regularity Of Solutions ◽  
Unified Framework ◽  
Nonlinear Water Waves ◽  
Eulerian Formulation ◽  
The Cauchy Problem ◽  
Dimensional Gravity

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity–capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals GLOBAL REGULARITY TO THE NAVIER-STOKES EQUATIONS FOR A CLASS OF LARGE INITIAL DATA

2018 ◽  
Vol 23 (2) ◽  
pp. 262-286
Author(s):  
Bin Han ◽  
Yukang Chen
Keyword(s):  
Initial Data ◽  
Initial Velocity ◽  
Global Regularity ◽  
Stokes Equations ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Large Initial Data ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Well Posed

In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3): ∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Study on the Interaction of Nonlinear Water Waves considering Random Seas

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Marten Hollm ◽  
Leo Dostal ◽  
Hendrik Fischer ◽  
Robert Seifried
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Random Seas

Meshless numerical simulation for fully nonlinear water waves

2005 ◽  
Vol 50 (2) ◽  
pp. 219-234 ◽  
Author(s):  
Nan-Jing Wu ◽  
Ting-Kuei Tsay ◽  
D. L. Young
Keyword(s):  
Numerical Simulation ◽  
Water Waves ◽  
Nonlinear Water Waves ◽  
Fully Nonlinear

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

scholarly journals Nonlinear water waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1501-1504 ◽  
Author(s):  
Adrian Constantin
Keyword(s):  
Water Waves ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Recent Developments

This introduction to the issue provides a review of some recent developments in the study of water waves. The content and contributions of the papers that make up this Theme Issue are also discussed.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

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