Related papers: Well-posedness in H^1 for the (generalized) Benjam…

We consider the Benjamin-Ono equation in the spatially quasiperiodic setting. We establish local well-posedness of the initial value problem with initial data in quasiperiodic Sobolev spaces. This requires developing some of the fundamental…

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence…

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^\alpha u\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally well-posed in the…

We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well-posedness results in weighted Sobolev spaces via contraction principle under minimal requirements in the…

We prove the discontinuity for the weak $ L^2(\T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(\T) $ as soon as $ s<0 $ and thus completes exactly the…

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 Kenig's result, which moreover provides…

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 critical Sobolev exponent $s_c=-1/2$ of…

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $…

We prove that the Benjamin Ono equation is globally well-posed in $H^s(\mathbb{R})$ for $s > 1/2$. Our approach does not rely on the global gauge transformation introduced by Tao (arXiv:math/0307289). Instead, we employ a modified version…

We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique…

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $$ \partial_tv-b\mathcal{H}\partial^2_xv- a\epsilon \partial_x^3v=cv\partial_xv-d\epsilon…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq…

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data $\phi$ belongs to $H^s$ for $s\geq 1/2$ with $\norm{\phi}_{L^2}$ sufficiently small without performing a gauge transformation.…

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also…

We study the Cauchy problem for the dissipative Benjamin-Ono equations $u_t+\H u_{xx}+|D|^\alpha u+uu_x=0$ with $0\leq\alpha\leq 2$. When $0\leq\alpha< 1$, we show the ill-posedness in $H^s(\R)$, $s\in\R$, in the sense that the flow map…

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 $H^\frac 1 2(\R)$ data, which is the natural ``finite energy'' class. Moreover,…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<\alpha \leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array}…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost…

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these…