Related papers: Strong Ill-Posedness in $L^\infty$ for the Riesz T…

In this paper, authors show the ill-posedness of 3D incompressible Navier-Stokes equations in the critical Triebel-Lizorkin spaces $ \dot{F}^{-1,q}_{\infty} (\mathbb{R}^3) $ for any $ q>2 $ in the sense that arbitrarily small initial data…

We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local…

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl…

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund…

We consider the Cauchy problem for the nonlinear Schr\"odinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS…

This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right.…

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution…

In this note, we study the ill-posedness problem for the derivative nonlinear Schr\"odinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm…

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of…

We study the initial value problem of the quadratic nonlinear Schr\"odinger equation $$ iu_t+u_{xx}=u\bar{u}, $$ where $u:\R\times \R\to \C$. We prove that it's locally well-posed in $H^s(\R)$ when $s\geq -\dfrac{1}{4}$ and ill-posed when…

In this paper, we consider the Cauchy problem to the Ericksen-Leslie system of liquid crystals in $\mathbb R^3$. Global well-posedness of strong solutions are obtained under the condition that the product of $\|u_0\|_2+\|\nabla d_0\|_2$ and…

In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$…

We study global well-posedness of strong solutions for the nonhomogeneous Navier-Stokes equations with density-dependent viscosity and initial density allowing vanish in $\mathbb{R}^2$. Applying a logarithmic interpolation inequality and…

We consider a complexification of the Euler equations introduced by \v{S}ver\'ak which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

In this paper, we consider the high-Weissenberg number limit of a Voigt-regularized two-dimensional Oldroyd-B model for viscoelastic fluids. We first demonstrate that the Euler-Oldroyd-B system is both linearly and nonlinearly ill-posed in…

We study the low regularity well-posedness for Cauchy problem of 3D relativistic Euler equations. Firstly, we introduce a new decomposition for relativistic velocity and derive new transport equations for vorticity, which both play a…

In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the local…

In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic…