Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the local analyticity for the Euler equations

Igor Kukavica, Maria Carmela Lombardo, Marco Sammartino

TL;DR

This work studies the Euler equations on a half-space with initial data analytic near the boundary and Sobolev regular away from it, introducing the diamond-analyticity framework to capture spatial decay of analyticity. The authors develop analytic-Sobolev function spaces and a nonstandard Cauchy–Kowalevskaya-type iteration, combined with a Leray projection and a novel mollification, to prove existence and uniqueness of solutions while allowing arbitrary initial quotients between imaginary and real analyticity radii. A linearized problem is solved using an analytic regularization with uniform estimates, then extended to the nonlinear problem via contraction arguments in time-weighted norms that shrink the analytic domain. The results reveal that diamond-analyticity can persist in a controlled diamond-shaped region, offering new insights into boundary effects and potential inviscid-limit behavior for fluid equations, with implications for complex singularities and boundary-layer interactions.

Abstract

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.

On the local analyticity for the Euler equations

TL;DR

This work studies the Euler equations on a half-space with initial data analytic near the boundary and Sobolev regular away from it, introducing the diamond-analyticity framework to capture spatial decay of analyticity. The authors develop analytic-Sobolev function spaces and a nonstandard Cauchy–Kowalevskaya-type iteration, combined with a Leray projection and a novel mollification, to prove existence and uniqueness of solutions while allowing arbitrary initial quotients between imaginary and real analyticity radii. A linearized problem is solved using an analytic regularization with uniform estimates, then extended to the nonlinear problem via contraction arguments in time-weighted norms that shrink the analytic domain. The results reveal that diamond-analyticity can persist in a controlled diamond-shaped region, offering new insights into boundary effects and potential inviscid-limit behavior for fluid equations, with implications for complex singularities and boundary-layer interactions.

Abstract

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.

Paper Structure

This paper contains 29 sections, 24 theorems, 145 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The set up and the main result
  3. Function spaces
  4. Analyticity close to the boundary: analytic norms and function spaces
  5. Analyticity on the half-space: analytic norms and function spaces
  6. Sobolev norms
  7. The weighted norm
  8. The combined norms
  9. Norms with a stopping time
  10. The analytic estimates
  11. The projection operator
  12. The linearized problem
  13. The regularized version: existence and uniqueness of a solution analytic in the conoid, with a large norm
  14. The regularized version: uniform in $\varepsilon$ estimates of the regularized solution in the diamond norm
  15. Euler equations
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let $0<\theta_0<\pi/2$ and $m\geq 3$. Suppose that $\hbox{\boldmath$u$\unboldmath}_{\text{in}}\in H^m_{\theta,\text{D}}\cap H^m_{\text{a}}$. Then there exists $\beta>0$ such that the system EQ12--EQ14 admits a unique solution $\hbox{\boldmath$u$\unboldmath}\in H^m_{\theta-\beta t,\text{D}}\cap H^m_a

Figures (2)

  • Figure 1: Domain of analyticity for functions analytic close to the boundary.
  • Figure 2: Domain of analyticity for functions analytic on the half-plane.

Theorems & Definitions (35)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Theorem 5.1
  • ...and 25 more