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On the conservation laws and the structure of the nonlinearity for SQG and its generalizations

Philip Isett, Andrew Ma

Abstract

Using a new definition for the nonlinear term, we prove that all weak solutions to the SQG equation (and mSQG) conserve the angular momentum. This result is new for the weak solutions of [Resnick, '95] and rules out the possibility of anomalous dissipation of angular momentum. We also prove conservation of the Hamiltonian under conjecturally optimal assumptions, sharpening a well-known criterion of [Cheskidov-Constantin-Friedlander-Shvydkoy, '08]. Moreover, we show that our new estimate for the nonlinearity is optimal and that it characterizes the mSQG nonlinearity uniquely among active scalar nonlinearities with a scaling symmetry.

On the conservation laws and the structure of the nonlinearity for SQG and its generalizations

Abstract

Using a new definition for the nonlinear term, we prove that all weak solutions to the SQG equation (and mSQG) conserve the angular momentum. This result is new for the weak solutions of [Resnick, '95] and rules out the possibility of anomalous dissipation of angular momentum. We also prove conservation of the Hamiltonian under conjecturally optimal assumptions, sharpening a well-known criterion of [Cheskidov-Constantin-Friedlander-Shvydkoy, '08]. Moreover, we show that our new estimate for the nonlinearity is optimal and that it characterizes the mSQG nonlinearity uniquely among active scalar nonlinearities with a scaling symmetry.
Paper Structure (30 sections, 19 theorems, 162 equations)

This paper contains 30 sections, 19 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Optimality of the definition
  3. Organization of the paper
  4. Acknowledgments
  5. Definitions of basic function spaces
  6. Hamiltonian conservation: The test function step
  7. The test function step in Euclidean space
  8. Conservation of the Hamiltonian
  9. Hamiltonian conservation in the Euclidean case
  10. Deriving the bilinear form
  11. Defining the nonlinearity
  12. The nonperiodic case
  13. Remark on generalized estimates of this type
  14. Continuous extension of the operator
  15. Extension to the homogeneous Sobolev space: Definition, boundedness and uniqueness
  16. ...and 15 more sections

Key Result

Theorem 1

Every weak solution to the mSQG equation ($0 \leq \delta \leq 1/2$) (of class $L_t^2 \dot{H}^{-1+\delta}(I \times \mathbb{R}^2)$) conserves the mean, impulse and angular momentum (defined in a generalized sense).

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2: Onsager singularity theorem
  • Theorem 3
  • Theorem 4: mSQG bound
  • Theorem 5
  • Theorem 6: Characterization of the nonlinearity
  • Definition 1.1
  • Definition 1.2
  • Theorem 7: Conservation laws
  • Definition 1.3
  • ...and 16 more