Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations

Dongho Chae, In-Jee Jeong, Jungkyoung Na, Sung-Jin Oh

TL;DR

This work analyzes the Ohkitani logarithmic SQG model, proving local well-posedness in Sobolev spaces with a time-dependent exponent that decreases in time, thereby circumventing ill-posedness in fixed spaces. It also shows that the δ-SQG family converges to the Ohkitani model under time rescaling, with a quantifiable rate and maximal existence time growing like $δ^{-1}$ as $δ\to0$, and establishes global well-posedness for small δ under logarithmic dissipation, including a local well-posedness result when the dissipation dominates the log. The main technique is a losing estimate in a scale of time-dependent Sobolev spaces built from a multiplier evolving along characteristics, linking the δ- and log-SQG dynamics and clarifying the borderline nature of the model. Together with the previous CCCGW results, these findings illuminate the long-time dynamics and global behavior of a class of generalized SQG equations with near-critical dissipation and singular velocity fields.

Abstract

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t θ- \nabla^\perp \log(10+(-Δ)^{\frac12})θ\cdot \nabla θ= 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the $δ$-SQG equations, defined by $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ= 0,$$ for all sufficiently small $δ>0$ depending on the size of the initial data. For the same range of $δ$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ+ \log(10+(-Δ)^{\frac12})θ= 0.$$

Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations

TL;DR

This work analyzes the Ohkitani logarithmic SQG model, proving local well-posedness in Sobolev spaces with a time-dependent exponent that decreases in time, thereby circumventing ill-posedness in fixed spaces. It also shows that the δ-SQG family converges to the Ohkitani model under time rescaling, with a quantifiable rate and maximal existence time growing like as , and establishes global well-posedness for small δ under logarithmic dissipation, including a local well-posedness result when the dissipation dominates the log. The main technique is a losing estimate in a scale of time-dependent Sobolev spaces built from a multiplier evolving along characteristics, linking the δ- and log-SQG dynamics and clarifying the borderline nature of the model. Together with the previous CCCGW results, these findings illuminate the long-time dynamics and global behavior of a class of generalized SQG equations with near-critical dissipation and singular velocity fields.

Abstract

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the -SQG equations, defined by for all sufficiently small depending on the size of the initial data. For the same range of , we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart:
Paper Structure (8 sections, 9 theorems, 142 equations)

This paper contains 8 sections, 9 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous works
  4. Ideas of the proof
  5. Open problems
  6. Proofs
  7. Local well-posedness for \ref{['eq:sqg-log']} and \ref{['eq:sqg-log-diss']}
  8. The relation between solutions of \ref{['eq:sqg-delta']} and \ref{['eq:sqg-log']}

Key Result

Theorem A

For any $s_0>4$ and $\theta_0\in H^{s_0}(\Omega)$, there exist $T = T(s_0,\Vert{\theta_0}\Vert_{H^{s_0}})>0$, a continuous decreasing function $s(t)>4$ with $s(0)=s_0$ defined in $t\in[0,T]$, and a solution $\theta\in L^\infty([0,T];H^{s(t)}(\Omega))$ to eq:sqg-log with initial data $\theta_0$ satis The solution is unique in the class $L^\infty([0,T];H^4(\Omega))$.

Theorems & Definitions (20)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Theorem D
  • Remark 1.3
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:wp-log']}
  • ...and 10 more