标题： 关于欧拉方程的局部解析性 显示英文标题

标题： On the local analyticity for the Euler equations

摘要： 在本文中，我们研究了具有初始条件的欧拉方程解的存在性和唯一性，这些初始条件在边界附近表现出解析正则性，而在远离边界处表现出Sobolev正则性。 这项工作的关键贡献是引入了钻石解析性框架，该框架以结构化的方式捕捉解析性半径的空间衰减，优于均匀解析性方法。 我们采用Leray投影和一种非标准的平滑技术来证明解析性半径的虚部与实部之间的商不受限制，从而将解析性持续结果扩展到传统约束之外。 我们的方法结合了解析-Sobolev估计与一种在Cauchy-Kowalevskaya框架中非标准的迭代方案，确保对解的演变进行严格控制。 这些结果有助于更深入地理解解析性与边界效应在流体方程中的相互作用。 它们可能对无粘性极限的纳维-斯托克斯方程的研究以及复奇点在流体动力学中的作用产生影响。 显示英文摘要

摘要： In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.