Interaccion Persona-ordenador

Curso 2026-2027

For the packet estimate, retain these smooth data and assume §3.2 (i)–(iii).

Proposition 3.1. There is a finite exponent $Q=Q(c_{0})$, uniform over the smooth packet data satisfying §3.2 (i)–(iii), such that for every $k\geq k_{0}(c_{0})$ satisfying

there is a smooth odd exact Euler solution $u^{\text{new}}$ on $[0,S]$, with pressure $p^{\text{new}}$.

1. 1.
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   Gradient and pressure increments. At $x=X(t,\ell y)$, with $\theta=km_{0}\cdot y$, its increments satisfy

   equation (3.13) (3.13)

   $\displaystyle\nabla u^{\text{new}}-\nabla u$

   $\displaystyle{}=a\chi_{1}v\otimes mf_{\delta}^{\prime}(\theta)+O(k^{-1/4}),$

   equation (3.14) (3.14)

   $\displaystyle\nabla^{2}p^{\text{new}}-\nabla^{2}\bar{p}$

   $\displaystyle{}=-2a\chi_{1}(m\cdot Mv)\frac{m\otimes m}{D_{m}}f_{\delta}^{\prime}(\theta)+O(k^{-1/4})\mathchar 46\relax$

   The errors are uniform globally in space and on $[0,S]$.

2. 2.
   ​

   Particle-map bounds. Suppose in addition that, for every integer $n\geq 0$, the parent particle map obeys

   $$\sum_{|I|=n}\sup_{0\leq t\leq S}\|\partial_{a}^{I}(X-\text{id},X_{t},X_{tt})\|_{H^{s}}\leq K_{h}^{n+1}(n!)^{2},\quad 1\leq K_{h}\leq P^{c_{0}}\mathchar 46\relax$$

   Then, for every integer $n\geq 0$, the particle map $X^{\text{new}}$ of $u^{\text{new}}$ obeys

   $$\sum_{|I|=n}\sup_{0\leq t\leq S}\|\partial_{a}^{I}(X^{\text{new}}-\text{id},X^{\text{new}}_{t},X^{\text{new}}_{tt})\|_{H^{s}}\leq(k^{C_{*}})^{n+1}(n!)^{2},\quad C_{*}=10(s+2)\mathchar 46\relax$$
3. 3.
   ​

   Initial increment. The initial increment splits as $u_{\text{high},0}+u_{\text{mean},0}$, with supports in $\{|a|\leq\ell/2\}$ and $\{|a|\leq 2\}$, respectively. For each fixed integer $m\geq 0$,

   $$\|u_{\text{high},0}\|_{H^{m}}\leq\ell^{-m}ak^{m}P^{c_{m}},\qquad\|u_{\text{mean},0}\|_{H^{m}}\leq\ell^{-m}k^{-2}P^{c_{m}}\mathchar 46\relax$$

   If $t_{0}=0$ and $L=0$, then $u_{\text{mean},0}=0$. Let $\phi_{\delta}$ be the mean-zero periodic primitive of $f_{\delta}$. In this case, the initial oscillatory increment is

   $$u_{\text{high},0}(a)=-\frac{\ell^{2}a}{k^{2}}\text{curl}_{a}\left[\chi_{1}(a/\ell)(m_{0}\times v(0))\phi_{\delta}(km_{0}\cdot a/\ell)\right]\mathchar 46\relax$$

Proof strategy for Proposition 3.1. We first solve the linear equation for the angle…