Related papers: Unconditional uniqueness for the Benjamin-Ono equa…
Using the explicit formula of P. G\'erard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle $\mathbb{T}= \mathbb{R}/2\pi\mathbb{Z}$ with bounded initial data $u_0\in…
A spectral analysis is done on the $L$ operator of the Lax pair for the Benjamin-Ono equation. Simplicity and finiteness of the discrete spectrum are established as are needed for the Fokas and Ablowitz inverse scattering transform scheme.…
We construct global-in-time singular dynamics for the (renormalized) cubic fourth order nonlinear Schr\"odinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the…
We consider the Cauchy problem for the logarithmic Schr\"odinger equation and prove uniqueness of weak $H^s(\mathbb{R}^d)$ solutions for $s\in(0,1)$, which improves on the previous uniqueness result in $H^1(\mathbb{R}^d)$. The proof is…
We show that uniqueness results of the kind those obtained for KdV and Schr\"odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces $$H^s(\R)\cap L^2(x^{2r}dx),$$ for…
The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile…
We consider solutions to the Benjamin-Ono equation $$\partial_t u - H \partial_x^2 u = -\partial_x(u^2)$$ that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least…
We study existence and uniqueness of solutions for second order ordinary stochastic differential equations with Dirichlet boundary conditions on a given interval. In the first part of the paper we provide sufficient conditions to ensure…
In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which…
By means of Banach fixed point theorem , the uniqueness of Boltzmann Equation generalizaled Solutions in Sobolev spaces in $L^1(\mathbb{R}^+\times \mathbb{R}^n\times \mathbb{R}^n)$, can be proved as well as Boltzmann Equation renormalized…
The Benjamin Ono equation with a slowly varying potential is $$ \text{(pBO)} \qquad u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton…
We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$.…
In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data…
We prove low regularity local well-posedness results in Bourgain-Klainerman-Machedon spaces for the Chern-Simons-Dirac system in the temporal gauge and the Coulomb gauge. Under slightly stronger assumptions on the data we also obtain…
We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the…
We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear…
We shall deduce some special regularity properties of solutions to the IVP associated to the Benjamin-Ono equation. Mainly, for datum $u_0\in H^{3/2}(\mathbb R)$ whose restriction belongs to $H^m((b,\infty))$ for some $m\in\mathbb…
We study persistence properties of solutions of the Benjamin-Ono equation in weighted Sobolev spaces. Roughly, we show that for $\beta<7/2$, the solution $u(x,t)$ of the BO remains in the space $L^2(|x|^{2\beta} dx)$ if and only if its data…
In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications.…
In this work we continue our study initiated in \cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in…