Related papers: Local well-posedness for third order Benjamin-Ono …

We prove the well-posedness of the Cauchy problem on torus to an eletromagnetoelastic system. The physical model consists of three coupled partial differential equations, one of them is a hyperbolic equation describing the elastic medium…

In this paper, we study the Cauchy problem for the 3D energy-critical inhomogeneous nonlinear Schr\"odinger equation(INLS) $$i\partial_{t}u+\Delta u=\pm|x|^{-\alpha}|u|^{4-2\alpha}u$$ with strong singularity $3/2\leq \alpha<2$. The…

In this manuscript, we would established in low regularity spaces $H^\ell, \ell\in [0,1)$, the existence and stability results of time-periodic solution of 1D Cauchy problem of forced damped Benjamin-Bona-Mahony equation (BBM). We use…

The Cauchy problem for a unified family of integrable $U(1)$-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an $L^1$…

In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on $u$ and $\partial_t u$. We prove a ill-posedness result for the "defocusing" case, and…

In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded…

We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poisson system in the energy class. Though the Newtonian potential diverges at the spatial infinity in the logarithmic order, global well-posedness is proven in both…

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity…

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 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,…

In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation by constructing some special spaces $E^s_{p,r}$, which…

We consider some parabolic equations which are model problems for a variety of nonlinear generalizations to the Black-Scholes equation of mathematical finance. In particular, we prove local well-posedness for the Cauchy problem with initial…

We establish the approximate controllability in $L^2$ for the nonlinear Benjamin-Ono equation on torus via two-dimensional control input. Our proof is based on adaptations of geometric control approach introduced by Agrachev and Sarychev.…

In this work we shall show that the Cauchy problem \begin{equation} \left\{ \begin{aligned} &(u_t+u^pu_x+\mathcal H\partial_x^2u+ \alpha\mathcal H\partial_y^2u )_x - \gamma u_{yy}=0 \quad p\in{\nat} &u(0;x,y)=\phi{(x,y)} \end{aligned}…

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are…

We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…

In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time)…

For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) +…

In this work we shall study the well-posedness and ill-posedness of the Cauchy problem associated to the equation \begin{equation*} u_{t}+a(u^{n})_{x}+(b\mathscr{H} u_{t}+u_{yy})_{x}=0, \end{equation*} in anisotropic weigthed Sobolev…

We discuss the existence of eigenvalues for a third order boundary value problem subject to functional boundary conditions and higher order derivative dependence in the nonlinearities. We prove the existence of positive and negative…