Related Experiment Video
Updated: Feb 17, 2026
04:35
Preparation of Free-Surface Hyperbolic Water Vortices
Published on: July 28, 2023
3.8K
Recent advances on the global regularity for irrotational water waves
1Department of Mathematics, Princeton University, Princeton, NJ, USA aionescu@math.princeton.edu.
Summary
This review covers long-time regularity for water waves equations. It details global existence and regularity results for gravity and gravity-capillary water waves, advancing the understanding of fluid dynamics.
Area of Science:
- Fluid dynamics
- Partial differential equations
- Mathematical analysis
Background:
- The Cauchy problem for water waves equations presents challenges in understanding long-time solution behavior.
- Free boundary Euler equations govern fluid motion but require advanced analytical techniques for regularity analysis.
Purpose of the Study:
- To review recent advancements in the long-time regularity of solutions to the water waves equations in 2D and 3D.
- To present a unified framework for global existence and regularity results.
Main Methods:
- Introduction of the free boundary Euler equations.
- Application of the paradifferential approach for local existence.
- Utilizing the Eulerian formulation for global regularity analysis.
Main Results:
- Global existence results for 3D and 2D gravity water waves.
- Global regularity for the 3D gravity-capillary water waves model (joint work with Deng and Pausader).
Conclusions:
- The study consolidates progress in understanding the long-time behavior and regularity of water waves solutions.
- Discussion on singularity formation and pointers to related research areas in water wave theory.
Related Concept Videos
01:07Propagation of Waves
3.1K
When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second medium. In such a case, the interface of the two mediums can be considered as a boundary that is neither fixed nor free.
Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...
Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...
3.1K
01:28Irrotational Flow
1.0K
Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:
1.0K
01:02Equations of Wave Motion
8.6K
Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
8.6K
01:04Travelling Waves
7.1K
A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. Mechanical waves are governed by Newton's laws and require a medium to travel. A medium is a substance in which a mechanical wave propagates, and the medium produces an elastic restoring force when it is deformed.
Water waves, sound waves, and seismic waves are some examples of mechanical waves. For water waves, the wave propagation medium is...
Water waves, sound waves, and seismic waves are some examples of mechanical waves. For water waves, the wave propagation medium is...
7.1K
01:22Surface Tension of Fluid
1.7K
Surface tension is a fundamental property of fluids, occurring at the boundary between a liquid and a gas or between two immiscible liquids. This phenomenon arises from the cohesive forces between molecules at the fluid's surface, creating an effect similar to a stretched elastic membrane. Inside each fluid, molecules are equally attracted in all directions by neighboring molecules, but surface molecules experience a net inward force, resulting in surface tension.
Surface tension varies...
Surface tension varies...
1.7K
01:06Divergence and Stokes' Theorems
3.8K
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
3.8K