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We study the initial value problem for the wave maps defined on the cyclic spacetime with the target Riemannian manifold that is responsive (see definition of the self coherence structure) to the parametric resonance phenomena. In particular, for arbitrary small and smooth initial data we construct blowing up solutions of the wave map if the metric of the base manifold is periodic in time.
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We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision gain term in Boltzmanns equation, combined with a novel application of the Kaniel-Shinbrot iteration.
We consider co-rotational wave maps from Minkowski space in $d+1$ dimensions to the $d$-sphere. Recently, Bizon and Biernat found explicit self-similar solutions for each dimension $dgeq 4$. We give a rigorous proof for the mode stability of these self-similar wave maps.
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iomega t} Q_v(x-vt) $$ with frequency $omega in R$, velocity $v in R^d$, and some finite-energy profile $Q_v in H^{1/2}(R^d)$, $Q_v ot equiv 0$. We prove that traveling solitary waves for speeds $|v| geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $sqrt{-DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d geq 2$.
We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $dgeq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on a method developed by the second author and Schorkhuber, we prove the asymptotic nonlinear stability of the ground-state self-similar solution.
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 solitary waves
and a Lax pair structure together with its implications (e.,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.
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