论文标题
一类二阶非线性双曲线方程的爆炸：非线性牛仔裤不稳定性模型
Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability
论文作者
论文摘要
了解宇宙和恒星系统中非线性结构的形成至关重要。非线性牛仔裤不稳定性在这些形成过程中起关键作用。一个多世纪以来，这一直是天体物理学的长期开放问题。在本文中，我们着重于牛顿宇宙中非线性牛仔裤不稳定性的简化模型，该模型由一类二阶非线性双曲线方程描述。 \ begin {equation*} \ box \ varrho（x^μ） +\ frac {\ mathcal {a}} {t} {t} \ partial_ {t} \ varrho（x^μ） - \ frac {\ Mathcal {b}} {t^2} \ varrho（x^μ）（1+ \ varrho（x^μ）） - \ frac {\ mathcal {c} - \ Mathcal {k Mathcal {k}}} （\ partial_ {t} \ varrho（x^μ））^2 = \ Mathcal {k} f（t）。 \ end {方程*}我们为此方程式建立了一个非线性自我爆炸解决方案的家族（解决方案本身在稳定的ode型爆炸中变得无限）。此外，我们提供了$ \ varrho $的增长率的估计，这可能有助于解释为什么宇宙中的非线性结构在天体物理观察中的生长速度要比经典牛仔裤不稳定的速度快得多。
Understanding the formation of nonlinear structures in the universe and stellar systems is crucial. The nonlinear Jeans instability plays a key role in these formation processes. It has been a long-standing open problem in astrophysics for more than a century. In this article, we focus on a reduced model of the nonlinear Jeans instability in an expanding Newtonian universe, which is described by a class of second-order nonlinear hyperbolic equations. \begin{equation*} \Box \varrho(x^μ) +\frac{\mathcal{a} }{t} \partial_{t}\varrho(x^μ) - \frac{\mathcal{b}}{t^2} \varrho(x^μ) (1+ \varrho(x^μ) ) -\frac{\mathcal{c}-\mathcal{k} }{1+\varrho(x^μ)} (\partial_{t}\varrho(x^μ))^2= \mathcal{k} F(t). \end{equation*} We establish a family of nonlinear self-increasing blowup solutions (where the solution itself becomes infinite in a stable ODE-type blowup) for this equation. Furthermore, we provide estimates on the growth rate of $\varrho$, which may help explain why the nonlinear structures in the universe grow much faster in astrophysical observations than predicted by the classical Jeans instability.
