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The singularity at degenerate points in steady axisymmetric compressible free surface flows with gravity

Lili Du, Chunlei Yang

Abstract

In this paper, we analyze the singular shape of the free boundary at degenerate points in a three dimensional axisymmetric compressible gravity flow. For all possible degenerate points on the free surface, we prove that the only nontrivial asymptotic behavior of the free surface at the stagnation points away from the axis of symmetry is the Stokes corner flow. The possible geometries for free boundaries at the non-stagnation axis points are downward pointing or upward pointing cusps. At the origin, there are only two nontrivial asymptotics possible: the Garabedian's pointed bubble or a horizontal flat surface. The problem is associated with the analysis of the degenerate points of a quasilinear free boundary problem of the Bernoulli type, and the main obstacles are the absence of a Weiss-type monotonicity formula. To achieve our goal, we establish for the first time monotonicity formulas for quasilinear Bernoulli type free boundary problems. Our formula works both when the equation becomes singular and when the free boundary condition is degenerate. Moreover, we establish a new nonlinear frequency formula at the horizontal flat points at the origin and integrate it with the compensated compactness theory for Euler equations, ensuring the strong convergence of variational solutions. Our results resolve the Stokes conjecture [Mathematical and Physical Papers, Vol. I., 1880] in a generalized compressible, three dimensional axisymmetric framework. In addition, it can also be realized as a compressible counterpart to Vǎrvǎrucǎ and Weiss [Comm. Pure Appl. Math., (67), 2014]. Our approach is completely new and gives a systematic approach for studying singularities of a singular Bernoulli type quasilinear free boundary problem.

The singularity at degenerate points in steady axisymmetric compressible free surface flows with gravity

Abstract

In this paper, we analyze the singular shape of the free boundary at degenerate points in a three dimensional axisymmetric compressible gravity flow. For all possible degenerate points on the free surface, we prove that the only nontrivial asymptotic behavior of the free surface at the stagnation points away from the axis of symmetry is the Stokes corner flow. The possible geometries for free boundaries at the non-stagnation axis points are downward pointing or upward pointing cusps. At the origin, there are only two nontrivial asymptotics possible: the Garabedian's pointed bubble or a horizontal flat surface. The problem is associated with the analysis of the degenerate points of a quasilinear free boundary problem of the Bernoulli type, and the main obstacles are the absence of a Weiss-type monotonicity formula. To achieve our goal, we establish for the first time monotonicity formulas for quasilinear Bernoulli type free boundary problems. Our formula works both when the equation becomes singular and when the free boundary condition is degenerate. Moreover, we establish a new nonlinear frequency formula at the horizontal flat points at the origin and integrate it with the compensated compactness theory for Euler equations, ensuring the strong convergence of variational solutions. Our results resolve the Stokes conjecture [Mathematical and Physical Papers, Vol. I., 1880] in a generalized compressible, three dimensional axisymmetric framework. In addition, it can also be realized as a compressible counterpart to Vǎrvǎrucǎ and Weiss [Comm. Pure Appl. Math., (67), 2014]. Our approach is completely new and gives a systematic approach for studying singularities of a singular Bernoulli type quasilinear free boundary problem.
Paper Structure (11 sections, 28 theorems, 314 equations, 10 figures)

This paper contains 11 sections, 28 theorems, 314 equations, 10 figures.

Table of Contents

  1. Introduction and setups
  2. Main Results
  3. Strategy of the proof
  4. Monotonicity formulas for axisymmetric compressible problems
  5. The blow-up analysis at the stagnation points
  6. The blow-up analysis at the non-stagnation axis points
  7. The blow-up analysis at the origin
  8. A nonlinear frequency formula at the origin
  9. Analysis of the frequency sequence
  10. Compensated compactness and the strong convergence
  11. The proof of the first variation formula

Key Result

Theorem 2.1

Let $u$ be a subsonic weak solution of fb, and assume that where $\Omega\subset\{(x_{1},x_{2}): x_{1} \geqslant 0\}$ so that $\Omega\cap\{x_{2}=0\}\neq\varnothing$. Then for any stagnation points $x^{\circ}=(x_{1}^{\circ},0)$ with $x_{1}^{\circ}\neq 0$ and any blow-up sequence The following statements hold:

Figures (10)

  • Figure 1: Axially symmetric compressible jet rising from a nozzle
  • Figure 2: The trichotomy principle for stagnation point
  • Figure 3: Classification of stagnation points
  • Figure 4: The trichotomy principal for non-stagnation axis points
  • Figure 5: Classification of non-stagnation axis point
  • ...and 5 more figures

Theorems & Definitions (63)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3: Subsonic variational solution
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9: Subsonic weak solutions.
  • Remark 1.10
  • ...and 53 more