We consider the Cauchy problem \[ \partial_t u+u\partial_x u+L(\partial_x u) =0, \quad u(0,x)=u_0(x) \] for a class of Fourier multiplier operators $L$, and prove that the solution map $u_0\mapsto u(t)$ is not uniformly continuous in $H^s$ on the real line or on the torus for $s > \frac{3}{2}$. Under certain assumptions, the result also hold for $s > 0$. The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries equations and we show that, in general, the flow map cannot be uniformly continuous if the dispersion of $L$ is weaker than that of the KdV operator. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant.

We prove that the Yang-Mills equation in Lorenz gauge in the (n+1)-dimensional case, $n \ge 4$, is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$, where $s > \frac{n}{2}-\frac{7}{8}$, $r > \frac{n}{2}-\frac{7}{4}$, respectively. The proof is based on the fundamental results of Klainerman-Selberg [7] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [16] and Tesfahun [19].

Knot energies, one of which is the Möbius energy, are constructed to measure the well-proportioned of the knot. The best-proportioned knot in the given knot class may be determined by the gradient flow of the energy. Indeed, Blatt showed the global existence and convergence of the gradient flow of the Möbius energy near stationary points. The Łojasiewicz inequality played an important role in proving this result. The inequality derived from $L^2$ representation of the first and second variations. On the other hand, Ishizeki and Nagasawa showed that the Möbius energy can be decomposed into three parts that are Möbius invariant. Each part has an $L^2$ representation of the first variation. In this paper, we discuss the $L^2$ representation of the second variation for each decomposed part of the Möbius energy, and derive explicit formulas for it. As a consequence of this and Chill's findings the Łojasiewicz inequality is derived.