Related papers: Low regularity well-posedness for KP-I equations: …
In this paper, low regularity local well-posedness results for the Kadomtsev--Petviashvili--I equation posed in spatial dimension $d =3$ are proved. Periodic, non-periodic and mixed settings as well as generalized dispersion relations are…
We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this…
We consider the well-posedness of the family of dispersion generalized Benjamin-Ono equations. Earlier work of Herr-Ionescu-Kenig-Koch established well-posedness with data in $L^2$, by using a discretized gauge transform in the setting of…
We show new well-posedness results in anisotropic Sobolev spaces for dispersion-generalized KP-I equations with increased dispersion compared to the KP-I equation. We obtain the sharp dispersion rate, below which generalized KP-I equations…
In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the…
We study global well-posedness for the Kadomtsev-Petviashvili II equation in three space dimensions with small initial data. The crucial points are new bilinear estimates and the definition of the function spaces. As by-product we obtain…
In this paper we obtain local in time existence and (suitable) uniqueness and continuous dependence for the KP-I equation for small data in the intersection of the energy space and a natural weighted $L^{2}$ space.
We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+\alpha\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $\alpha\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to…
We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in…
The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $\beta>1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric…
We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the…
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 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…
We consider the dispersion-generalized KP-II equation on a partially periodic domain in the weakly dispersive regime. We use Fourier decoupling techniques to derive essentially sharp Strichartz estimates. With these at hand, we show global…
New local well-posedness results for dispersion generalized Benjamin-Ono equations on the torus are proved. The family of equations under consideration links the Benjamin-Ono and Korteweg-de Vries equation. For sufficiently high dispersion…
We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high regularity for a class of nonlinear higher-order dispersive equations generalizing the KdV hierarchy both on the line and on the torus.
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…
In this paper, the local well-posedness of periodic fifth order dispersive equation with nonlinear term $P_1(u)\p_xu + P_2(u)\p_x u\p_xu $. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. We also get some new Strichartz estimates.
In this paper we study local well-posedness in the energy space for a family of dispersive equations that can be seen as dispersive ``interpolations'' between the KdV and the Benjamin-Ono equation.
In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves…