Hermite半群的相空间分析及其在非线性整体适定性中的应用
Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness
摘要：
研究了$\mathbb{R}^d$中的Hermite算子$H=-Δ+|x|^2$及其在相空间中的分数幂$H^β$，$β>0$.也就是说，我们通过所谓的短时傅里叶、别名傅里叶-维格纳或巴格曼变换$V_g f$（$g$是固定窗函数）来表示函数$f$，并且我们通过$V_g f$的相空间中的混合勒贝格范数来测量它们的规律性和衰减，即根据对调制空间$M^{p,q}$、$0< p,q\leq \infty$的隶属关系。我们证明了半群$e^{-tH^β}$作用于$M^{p,q}$时的固定时间估计的完整范围，对于每个$0< p,q\leq \infty$，表现出最佳的全局时间衰减以及相空间平滑。作为应用，我们建立了全局适定性的非线性热方程$H^β$与幂型非线性（聚焦或散焦），小的初始数据在调制空间或Wiener汞齐空间。我们表明，这样的全球解决方案具有相同的最佳衰减$e^{-c t}$作为相应的线性方程的解决方案，其中$c=d^β$是底部的$H^β$的频谱。这与无势的非线性聚焦热方程形成了鲜明的对比，在无势的非线性聚焦热方程中，对于（甚至很小的）常数初始数据，有限时间内的爆破总是发生，因此在$M^{\infty,1}$中也是如此。
英文摘要：
We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^β$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^β$ is the bottom of the spectrum of $H^β$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$.
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