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The inviscid inflow-outflow problem via analyticity

Igor Kukavica, Wojciech Ożański, Marco Sammartino

TL;DR

We address the inviscid incompressible Euler equations with inflow-outflow boundary data on an analytic domain via an analytic framework that uses a time-dependent analyticity radius $\tau(t)$ and tangential derivatives. The main contribution is local well-posedness in analytic spaces without compatibility conditions across all dimensions, with a comprehensive extension to general analytic domains through product and pressure estimates and a derivative-reduction lemma. In 2D, global well-posedness is established under decay of the inflow field $\overline{u}$, leveraging persistence of analyticity to control nonlinear growth and vorticity. The results provide a robust, compatibility-free approach to inflow-outflow Euler problems and illuminate the role of analyticity in accommodating derivative loss and boundary interactions.

Abstract

We consider the incompressible Euler equation on an analytic domain $Ω$ with nonhomogeneous boundary condition $u\cdot \mathsf{n} = \overline{u} \cdot \mathsf{n}$ on $\partial Ω$, where $\overline{u}$ is a given divergence-free analytic vector field. We establish local well-posedness for $u$ in analytic spaces without any compatibility conditions in all space dimensions. We also prove global well-posedness in the 2D case if $\overline{u}$ decays in time sufficiently fast.

The inviscid inflow-outflow problem via analyticity

TL;DR

We address the inviscid incompressible Euler equations with inflow-outflow boundary data on an analytic domain via an analytic framework that uses a time-dependent analyticity radius and tangential derivatives. The main contribution is local well-posedness in analytic spaces without compatibility conditions across all dimensions, with a comprehensive extension to general analytic domains through product and pressure estimates and a derivative-reduction lemma. In 2D, global well-posedness is established under decay of the inflow field , leveraging persistence of analyticity to control nonlinear growth and vorticity. The results provide a robust, compatibility-free approach to inflow-outflow Euler problems and illuminate the role of analyticity in accommodating derivative loss and boundary interactions.

Abstract

We consider the incompressible Euler equation on an analytic domain with nonhomogeneous boundary condition on , where is a given divergence-free analytic vector field. We establish local well-posedness for in analytic spaces without any compatibility conditions in all space dimensions. We also prove global well-posedness in the 2D case if decays in time sufficiently fast.
Paper Structure (12 sections, 11 theorems, 211 equations)

This paper contains 12 sections, 11 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Smoothness of solutions
  3. The case of a flat domain
  4. A priori analytic estimate (periodic channel)
  5. Proof of Theorem \ref{['T01']}
  6. The general case
  7. Preliminaries
  8. Proof of Theorem \ref{['T01']}
  9. Product estimate: the general case
  10. Pressure estimate
  11. Derivative reduction lemma
  12. Global well-posedness in $2$D

Key Result

Theorem 1.2

Let $r= d+1$. Assume that $\Omega \subset \mathbb{R}^d$, where $d\geq 2$, is an analytic domain, and suppose that $v_0\in X(\tau_0)$, for some $\tau_0\in(0,1]$. Then there exists $M\geq 1$ and a unique solution $v\in C ([0,T_0];\widetilde{X}(\tau(t)) )$ to intro_euler_v, satisfying the initial condi for all $t\in [0,T_0]$, where and $T_0\coloneqq \tau_0 /M$.

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2: Local well-posedness of \ref{['intro_euler_v']} in analytic spaces
  • Theorem 1.3: Global well-posedness in $2$D
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['L_product']}
  • Remark 2.2: The case of any dimension $d\geq 2$
  • Remark 2.3
  • Corollary 2.4: Related product estimates
  • proof
  • Lemma 2.5
  • ...and 13 more