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Register a new userGlobal wellposedness for 2D quasilinear wave without Lorentz
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We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.
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We consider the two-dimensional quasilinear wave equations with quadratic nonlinearities. We introduce a new class of null forms and prove uniform boundedness of the highest order norm of the solution for all time. This class of null forms include several prototypical strong null conditions as special cases. To handle the critical decay near the light cone we inflate the nonlinearity through a new normal form type transformation which is based on a deep cancelation between the tangential and normal derivatives with respect to the light cone. Our proof does not employ the Lorentz boost and can be generalized to systems with multiple speeds.
Hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $Box u = Q(u, u, u)$ where $Q$ vanishes to second order and $(partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $upartial_alpha u = frac{1}{2}partial_alpha u^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both results to the case of multiple speed systems.
In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in $H^{3}times H^{2}$. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $mathbb{R}timesmathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $gin L^{infty}(mathbb{R})$ is even with $g otequiv 0$ and $h(x)=gamma,delta_0(x)$ with $gammain{mathbb{R}}backslash{0}$ and $delta_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-frac{d^2}{dx^2}-k^2omega^2g$ on $L^2(mathbb{R})$ for all $kin 2mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitly given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.
This paper continues the development of regularity results for quasilinear measure data problems begin{align*} begin{cases} -mathrm{div}(A(x, abla u)) &= mu quad text{in} Omega, quad quad qquad u &=0 quad text{on} partial Omega, end{cases} end{align*} in Lorentz and Lorentz-Morrey spaces, where $Omega subset mathbb{R}^n$ ($n ge 2$), $mu$ is a finite Radon measure on $Omega$, and $A$ is a monotone Caratheodory vector valued operator acting between $W^{1,p}_0(Omega)$ and its dual $W^{-1,p}(Omega)$. It emphasizes that this paper studies the `very singular case $1<p le frac{3n-2}{2n-1}$ and the problem is considered under the weak assumption, where the $p$-capacity uniform thickness condition is imposed on the complement of domain $Omega$. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz-Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the `very singular case still remains a challenge, specifically related to Lorentz and Lorentz-Morrey spaces.
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