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Real analytic solutions to the divergence equation

Chi Hin Chan, Jun-Shuo Chen, Cheng-Fang Su

Abstract

In this paper, we develop a differential-topological method to yield explicit real analytic solutions $v$ to the divergence equation $div_{\mathbb{R}^n} v = f$ on any annali $A(R_1 ,R_2) = \{ x \in \mathbb{R}^n : R_1 < |x| < R_2\}$, with $n \geq 2$, and $0 < R_1 < R_2 < \infty$. The prescribed source term $f$ is supposed to be real analytic on $\overline{A(R_1 , R_2)} = \{ x \in \mathbb{R}^n : R_1 \leq |x| \leq R_2\}$ satisfying the zero integral condition on $A(R_1, R_2)$. The resulting solution $v$ is a real analytic vector field on $\overline{A(R_1 , R_2)}$, which vanishes on $\partial \big( A(R_1, R_2 ) \big )$. The method which we develop here is different from the standard Bogovski approach and the Kapitanskii-Pileckas approach. The first main step our method is a clever differential-topological argument, which we develop under the inspiration and guidance of the standard proof of the cohomological statement $H_c^n \big ( \mathbb{R}^n\big ) = \mathbb{R}$ in Spviak book A Comprehensive Introduction to Differential Geometry, Vol I. This allows us to reduce the problem to that of solving a linear algebra problem.

Real analytic solutions to the divergence equation

Abstract

In this paper, we develop a differential-topological method to yield explicit real analytic solutions to the divergence equation on any annali , with , and . The prescribed source term is supposed to be real analytic on satisfying the zero integral condition on . The resulting solution is a real analytic vector field on , which vanishes on . The method which we develop here is different from the standard Bogovski approach and the Kapitanskii-Pileckas approach. The first main step our method is a clever differential-topological argument, which we develop under the inspiration and guidance of the standard proof of the cohomological statement in Spviak book A Comprehensive Introduction to Differential Geometry, Vol I. This allows us to reduce the problem to that of solving a linear algebra problem.
Paper Structure (5 sections, 10 theorems, 137 equations)

This paper contains 5 sections, 10 theorems, 137 equations.

Table of Contents

  1. Introduction of our paper.
  2. Preparations
  3. The First Major Step
  4. The main Theorem of our paper.
  5. Appendix: comparison of Lemma 2.4 with Spivak proof of Theorem $9$ of Chapter $8$ in Spivak.

Key Result

Theorem 1.1

Consider $n \in \mathbb{Z}^+$ with $n \geq 2$, and $0< R_1 < R_2 < \infty$. Let $f : \overline{A(R_1, R_2)}\rightarrow \mathbb{R}$ be real analytic on $\overline{A(R_1 , R_2)}$ and satisfies $\int_{\overline{A(R_{1},R_2)}} f \mathop{\mathrm{Vol}}\nolimits_{\mathbb{R}^n} = 0$. Then, it follows that t holds for all $x \in \overline{A(R_1 , R_2 )}$, and that

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 10 more