# Low regularity conservation laws for the Benjamin–Ono equation

@article{Talbut2018LowRC, title={Low regularity conservation laws for the Benjamin–Ono equation}, author={Blaine Talbut}, journal={arXiv: Analysis of PDEs}, year={2018} }

We obtain conservation laws at negative regularity for the Benjamin-Ono equation on the line and on the circle. These conserved quantities control the $H^s$ norm of the solution for $-\frac{1}{2} < s < 0$. The conservation laws are obtained from a study of the perturbation determinant associated to the Lax pair of the equation.

## 12 Citations

Unconditional uniqueness for the Benjamin-Ono equation

- Mathematics
- 2021

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in H, both on the real line and on the torus. We use the gauge transformation of Tao and two…

A priori estimates for the derivative nonlinear Schrödinger equation

- Mathematics
- 2020

We prove low regularity a priori estimates for the derivative nonlinear Schrodinger equation in Besov spaces with positive regularity index conditional upon small $L^2$ -norm. This covers the full…

Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

- PhysicsLetters in Mathematical Physics
- 2021

Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrodingger equation (DNLS) with small mass in \cite{KlausS:DNLS}, we show the corresponding microscopic…

On the Benjamin-Ono equation on $\mathbb{T}$ and its periodic and quasiperiodic solutions

- Mathematics
- 2021

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions,…

A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications

- Mathematics
- 2020

The Benjamin-Ono equation was introduced by Benjamin in 1967 as a model for a special regime of internal gravity waves at the interface of two fluids. This nonlinear dispersive equation admits a Lax…

Growth Bound and Nonlinear Smoothing for the Periodic Derivative Nonlinear Schr\"odinger Equation

- Mathematics
- 2020

A polynomial-in-time growth bound is established for global Sobolev HpTq solutions to the derivative nonlinear Schrödinger equation on the circle with s ą 1. These bounds are derived as a consequence…

Polynomial bound and nonlinear smoothing for the Benjamin-Ono equation on the circle

- Mathematics
- 2020

A P ] 1 6 M ar 2 02 1 ON THE BENJAMIN – ONO EQUATION ON T AND ITS PERIODIC AND QUASIPERIODIC SOLUTIONS

- Mathematics
- 2021

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions,…

Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation

- MathematicsQuarterly of Applied Mathematics
- 2020

In this paper we characterize the Nazarov–Sklyanin hierarchy for the classical periodic Benjamin–Ono equation in two complementary degenerations: for the multiphase initial data (the periodic…

Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution

- Mathematics
- 2020

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the…

## References

SHOWING 1-10 OF 24 REFERENCES

The Benjamin—Ono equation in energy space

- Mathematics
- 2006

We prove existence of solutions for the Benjamin—Ono equation with data in H s(ℝ), s > 0. Thanks to conservation laws, this yields global solutions for H 1 2 (ℝ) data, which is the natural “finite…

Low regularity conservation laws for integrable PDE

- Mathematics
- 2017

We present a general method for obtaining conservation laws for integrable PDE at negative regularity and exhibit its application to KdV, NLS, and mKdV. Our method works uniformly for these problems…

Global well-posedness in the energy space for the Benjamin–Ono equation on the circle

- Mathematics
- 2005

AbstractWe prove that the Benjamin-Ono equation is locally well-posed in
$$H^{1/2}(\mathbb{T})$$. This leads to a global well-posedeness result in
$$H^{1/2}(\mathbb{T})$$ thanks to the energy…

Well-posedness results for the generalized Benjamin–Ono equation with small initial data

- Mathematics
- 2004

On the cauchy problem for the benjamin-ono equation

- Mathematics
- 1986

On discute l'existence, l'unicite et le comportement asymptotique (pour |x|→∞) des solutions du probleme a valeur initiale pour l'equation de Benjamin-Ono: δ t u=−δ x (u 2 +2σδ x u), u(x,o)=f(x)

GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H1(R)

- Mathematics
- 2004

We show that the Benjamin–Ono equation is globally well-posed in Hs(R) for s≥1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly…

Global well-posedness of the Benjamin–Ono equation in low-regularity spaces

- Mathematics
- 2005

We prove that the Benjamin-Ono initial value problem is globally well-posed in the Sobolev spaces $H^\sigma_r$, $\sigma\geq 0$.

On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations

- Mathematics
- 2003

We prove that the Benjamin-Ono equation is locally well-posed in Hs(R) for s > 9/8 and that for arbitrary initial data, the modified (cubic nonlinearity) Benjamin-Ono equation is locally well-posed…

The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited

- Mathematics
- 2010

In a recent work, Ionescu and Kenig proved that the Cauchy problem associatedto the Benjamin-Ono equation is well-posed in $L^2(\mathbb R)$. In this paper we give a simpler proof of Ionescu and…