AI 中文总结
研究从大初始数据构造二维不可压缩欧拉方程自相似解，通过耗散自相似剖面的消失耗散极限，关键在于临界洛伦兹空间中涡度剖面估计，得到局部能量有限等性质的欧拉解。
AI 中文摘要
设\(\frac{1}{3}<a<1\)，\(u_0\)是\(\mathbb{R}^2\setminus\{0\}\)上\(C^1\)、无散、\((-a)\)齐次向量场。构造二维不可压缩欧拉方程正向时间自相似解\(u(t,x)=t^{-\frac{a}{1+a}} U\left(\frac{x}{t^{{\frac{1}{1+a}}}}\right)\)，初始数据为\(u_0\)。对初始数据无小量或符号假设。通过耗散自相似剖面的消失耗散极限构造，关键是在临界洛伦兹空间\(L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)\)中对涡度剖面的估计，结果表明欧拉解\(u\)局部能量有限等。
英文摘要
Let $0<a<1$ and let $u_0$ be a $C^1$, divergence-free, $(-a)$-homogeneous vector field on $\mathbb{R}^2\setminus\{0\}$. We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, \[ u(t,x)=t^{-\frac{a}{1+a}} U\left(\frac{x}{t^{{\frac{1}{1+a}}}}\right), \] with initial datum $u_0$. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of hypodissipative self-similar profiles built directly in the critical vorticity space. The main estimate is a uniform critical Lorentz bound $\|\operatorname{curl} U\|_{L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)}$. The resulting Euler solution $u$ belongs to $C([0,\infty);L^2_{\mathrm{loc}}(\mathbb{R}^2))$ and has vorticity uniformly bounded in the critical space $L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)$. Its velocity converges strongly to $u_0$ in $L^2_{\mathrm{loc}}$, while its vorticity converges weak-star to $ω_0=\operatorname{curl} u_0$ in $L^{\frac{2}{1+a},\infty}$ as $t\downarrow0$.
Comments30 pages, enlarged the range of parameter $a$ by completely different construction of hypodissipative profile, previous one moved to Appendix B
