康斯坦丁 - 拉克斯 - 马伊达方程中自相似坍缩的谱图
The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation
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中文总结 AI 辅助
研究康斯坦丁 - 拉克斯 - 马伊达方程自相似坍缩剖面的谱图，通过围绕精确剖面线性化等方法，在$a = 0$时证明了本质谱、点谱等相关性质，对$a > 0$也有相关研究，给出了依赖于实现的坍缩剖面谱图。
中文摘要 AI 辅助
我们给出了康斯坦丁 - 拉克斯 - 马伊达（CLM）方程自相似坍缩剖面的谱描述，它是广义族$w_t + a\,u\,w_x = u_x\,w$，$u_x = Hw$中$a = 0$的锚点。围绕精确剖面$\Omega(y) = -y/(y^2 + 1/4)$进行线性化，并将$L_0$实现为原点 - $H^2$空间上的闭算子，我们在$a = 0$时证明了三件事。其一，其本质谱在单条垂直线$\{\mathrm{Re}\,\lambda = -1/2\}$处与闭半平面$\{\mathrm{Re}\,\lambda \geq -1/2\}$相交，该线由对数加宽的魏尔序列确定，明确的哈代 - 梅林预解式界建设性地清空了半平面除$0$和$1$之外的其余部分。其二，在奇数实现下，其在$\mathbb{C}$上的全点谱恰好是$\{0,1\}$，即缩放和时移对称模式，没有嵌入特征值；通过标准调制去除这些后，在$X$上留下$1/2$的谱隙。其三，线性半群及其精确衰减率$e^{-\tau/2}$以封闭形式计算，但在通过有界转移映射从$X$得到的共轭变量的加权空间上；由于$L_0$是非正规的且谱隙本身并不给出$X$范数下的衰减率，所以我们将两者分开。一种实现二分法确定了一般离散化的带内涂抹作为最大$L^2$实现的忠实谱，而原点 - $H^2$去除了该谱。对于$a > 0$，我们为每个可允许的光滑聚焦剖面证明了一个条件二线包含关系，重新计算了卢什尼科夫、西兰季耶夫和西格尔的分支$c_l(a)$作为交叉检查，并记录了形式缩放相关指数$s^*(a) = 1/c_l(a)$，对于固定的足够正则数据，在自相似变量中，低于该指数时分数耗散在渐近上是次主导的。贡献是坍缩剖面本身依赖于实现的谱图。
英文摘要
We give a spectral description of the self-similar collapse profile of the Constantin-Lax-Majda (CLM) equation, the $a=0$ anchor of the generalized family $w_t + a\,u\,w_x = u_x\,w$, $u_x = Hw$. Linearizing about the exact profile $Ω(y) = -y/(y^2+1/4)$ and realizing $L_0$ as a closed operator on the origin-$H^2$ space, we prove three things at $a=0$. Its essential spectrum meets the closed half-plane $\{\mathrm{Re}\,λ\ge -1/2\}$ in the single vertical line $\{\mathrm{Re}\,λ= -1/2\}$: the line is placed by a log-widening Weyl sequence, and an explicit Hardy-Mellin resolvent bound constructively empties the rest of the half-plane apart from $0$ and $1$. Its full point spectrum over $\mathbb{C}$, on the odd realization, is exactly $\{0,1\}$, the scaling and time-shift symmetry modes, with no embedded eigenvalues; removing these by the standard modulation leaves a spectral gap of $1/2$ on $X$. The linear semigroup and its exact decay rate $e^{-τ/2}$ are computed in closed form, but on a weighted space of the conjugated variable reached from $X$ by a bounded transfer map; we keep the two separate, since $L_0$ is non-normal and a spectral gap does not by itself give a decay rate in the $X$ norm. A realization dichotomy identifies the in-strip smear of generic discretizations as the faithful spectrum of the maximal $L^2$ realization, which origin-$H^2$ removes. For $a>0$ we prove a conditional two-line inclusion for each admissible smooth focusing profile, recompute the branch $c_l(a)$ of Lushnikov, Silantyev, and Siegel as a cross-check, and record the formal scaling-relevance exponent $s^*(a) = 1/c_l(a)$, below which fractional dissipation is asymptotically subdominant in self-similar variables for fixed sufficiently regular data. The contribution is the realization-dependent spectral picture of the collapse profile itself.
