Related papers: Ill-posedness for one-dimensional wave maps at the…

In this paper, we prove that the 2D viscous shallow water equations is ill-posed in the critical Besov spaces $\B^{\frac2p-1}_{p,1}(\R^2)$ with $p>4$. Our proof mainly depends on the method introduced by the paper \cite{C-M-Z4}.

In this article we initiate the study of 1+ 2 dimensional wave maps on a curved spacetime in the low regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical…

In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n \leq 5$. We show this equation, with power $2\le p\le 1+4/(n-1)$, is…

In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the…

In this paper we prove that the cubic wave equation is globally well - posed and scattering for radial initial data lying in $B_{1,1}^{2} \times B_{1,1}^{1}$. This space of functions is a scale invariant subspace of $\dot{H}^{1/2} \times…

We identify the wave maps type nonlinearities of incompressible Hookean elastodynamics equations in Lagerangian coordinates, and iterate them in the adapted $U^2$-type spaces to prove the small data global well-posedness in the critical…

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the…

We review the current state of results about the half-wave maps equation on the domain $\mathbb{R}^d$ with target $\mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling…

In this paper we prove global well-posedness for the defocusing, energy-subcritical, nonlinear wave equation on $\mathbb{R}^{1 + 3}$ with initial data in a critical Besov space. No radial symmetry assumption is needed.

We consider the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with $n \geqslant 3$, the well-posedness and ill-posedness in scaling critical spaces are…

We consider the stationary problem for the quasi-geostrophic equation on the whole plane and investigate its well-posedness and ill-posedness. In[Fujii, Ann. PDE 10, 10 (2024)], it was shown that the two-dimensional stationary…

In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that…

It is shown that both the Camassa-Holm and Novikov equations are ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with $(p,r)\in[1,\infty]\times(1,\infty]$ in \cite{Guo2019} and well-posed in $B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in…

We prove that the solution map associated with the $1D$ half-wave cubic equation in the periodic setting cannot be uniformly continuous on bounded sets of the periodic Sobolev spaces $H^s$ with $s\in (1/4, 1/2)$

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty,…

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions $n\geq4$, global wellposedness for data which is small in the critical…

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^2)$. As is known, this system is locally well-posed…

The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the…

In this paper, we study ill-posedness of cubic fractional nonlinear Schr\"odinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on $\mathbb R$. In particular, we prove the following ill-posedness results: (i)…

Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class…