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Shock formation for the 2D rotating shallow water equations with non-zero vorticity

Zhendong Chen, Chunjing Xie

TL;DR

This work proves finite-time shock formation for the two-dimensional rotating shallow water equations with non-zero vorticity by employing a geometric acoustical framework inspired by Christodoulou’s shock theory. The authors recast the system into covariant wave equations for density and velocity, coupled with a transport equation for the vorticity, and track the evolution using an acoustical metric $g$ and the inverse foliation density $μ$, whose vanishing signals shock formation. A rigorous, multi-layer energy-flux analysis is carried out, including a novel decent scheme to eliminate μ-weight degeneracies and a careful treatment of large vorticity via coupled wave-transport estimates; explicit short-pulse initial data are constructed to trigger the shock, and the shock time is computable (e.g., $T_*= frac{2}{3}+O(δ)$ under certain data). The results yield a complete geometric description of the shock mechanism in this setting, with first derivatives of $( ho,v^1)$ and the height blowing up while the potential vorticity remains Lipschitz, and they illuminate potential extensions to Euler-type systems and quasilinear Klein-Gordon equations in multiple dimensions.

Abstract

In the paper, the shock formation for the two-dimensional rotating shallow water system is established. We construct a large class of initial data which leads to the finite-time blow-up for the solutions. Moreover, the solutions are allowed to have non-zero large vorticity (in derivative sense), even up to the shock. Our results provide the first complete geometric description of the shock formation mechanism to the two-dimensional rotating shallow water system with vorticity. The formation of shock is characterized by the collapse of the characteristic hypersurfaces, where the first-order derivatives of the velocity, the height, and the specific vorticity blow up while the potential vorticity remains Lipschitz continuous. The methods developed in this paper should also be useful in studying the shock formation for the Euler equations with various source terms and a class of quasilinear Klein-Gordon equations in multi-dimensions.

Shock formation for the 2D rotating shallow water equations with non-zero vorticity

TL;DR

This work proves finite-time shock formation for the two-dimensional rotating shallow water equations with non-zero vorticity by employing a geometric acoustical framework inspired by Christodoulou’s shock theory. The authors recast the system into covariant wave equations for density and velocity, coupled with a transport equation for the vorticity, and track the evolution using an acoustical metric and the inverse foliation density , whose vanishing signals shock formation. A rigorous, multi-layer energy-flux analysis is carried out, including a novel decent scheme to eliminate μ-weight degeneracies and a careful treatment of large vorticity via coupled wave-transport estimates; explicit short-pulse initial data are constructed to trigger the shock, and the shock time is computable (e.g., under certain data). The results yield a complete geometric description of the shock mechanism in this setting, with first derivatives of and the height blowing up while the potential vorticity remains Lipschitz, and they illuminate potential extensions to Euler-type systems and quasilinear Klein-Gordon equations in multiple dimensions.

Abstract

In the paper, the shock formation for the two-dimensional rotating shallow water system is established. We construct a large class of initial data which leads to the finite-time blow-up for the solutions. Moreover, the solutions are allowed to have non-zero large vorticity (in derivative sense), even up to the shock. Our results provide the first complete geometric description of the shock formation mechanism to the two-dimensional rotating shallow water system with vorticity. The formation of shock is characterized by the collapse of the characteristic hypersurfaces, where the first-order derivatives of the velocity, the height, and the specific vorticity blow up while the potential vorticity remains Lipschitz continuous. The methods developed in this paper should also be useful in studying the shock formation for the Euler equations with various source terms and a class of quasilinear Klein-Gordon equations in multi-dimensions.

Paper Structure

This paper contains 43 sections, 29 theorems, 377 equations, 4 tables.

Table of Contents

  1. Introduction and Main Results
  2. Review of existing literature
  3. Shock formation to the Compressible Euler system
  4. Shock development for compressible Euler system
  5. Shock formation and global existence to the shallow water system
  6. Main Results
  7. Some key ingredients and features in this paper
  8. Review the main ideas in the existing literature on shock formation
  9. Review of main ideas in point shock formation to compressible Euler system
  10. Key ideas of the proof for the main results in this paper
  11. Explicit short pulse data
  12. The coupled energy estimates
  13. Large vorticity regime
  14. H$\ddot{\text{o}}$lder continuity for wave variables.
  15. Organization of the paper
  16. ...and 28 more sections

Key Result

Theorem 1.1

For the initial data constructed as in initialdata0-initialdata2 (see also Lemma initialdata), if the following condition holds: then the solution to the shallow water system shallowwater0 will form a shock at $t=T_{\ast}\leq 1$, which can be computed explicitly. Furthermore, $\partial_{\alpha}\rho$ and $\partial_{\alpha}v^1$ blow up at $T_{\ast}$ for some $\alpha\in\{0,1\}$ in the sense that fo

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • ...and 71 more