In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension $d\ge 2$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic $\mathcal C^\infty $ self-similar solutions with suitable decay at infinity. The $\mathcal C^\infty $ regularity is used in a fundamental way in our companion paper (part II) in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions $d=2,3$.

在本文及其后续文章中，我们构造了一组有限能量平滑初始数据，可压缩三维 Navier-Stokes 和 Euler 方程的相应解在稍后的某个时间点内爆（具有无限密度），并且我们完整地描述了奇点的相关形成。本文关注的是正压欧拉方程在空间无穷大处具有衰减密度的维度$d\ge 2$ 的平滑自相似轮廓的存在。Guderley 的开创性工作中介绍了控制球对称自相似解方程的非线性 ODE 的相图。它使我们能够构建自相似问题的全局轮廓，但是结果证明在相关的声锥上通常是不平滑的。量化速度累积到一个临界值，我们证明了非泛型 $\mathcal C^\infty $ 自相似解的存在，在无穷远处具有适当的衰减。$\mathcal C^\infty $ 正则性在我们的配套论文（第二部分）中以基本方式用于分析相关的线性化算子，并导致构建可压缩的有限能量爆炸解维度 $d=2,3$ 的 Euler 和 Navier-Stokes 方程。