Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations

Abstract

We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.

Appendix A: Bookkeeping of Various Parameters

In this appendix we sketch how the choice of various parameters in (1.4) take effect on various error terms and the regularity of the weak solution. Recall that (observe from below $\log \mu _{n+1} \sim \log \lambda _n$)

$$\begin{aligned} \lambda _n=\left\lceil \lambda _0^{b^n} \right\rceil ,\quad r_n=\lambda _n^{-\beta }, \quad \mu _{n+1}=(\lambda _n \lambda _{n+1})^{\frac{1}{2}}, \quad {\alpha = \frac{1}{2} + \frac{\beta }{2b} -(b-1)^3>\frac{1}{2}}. \end{aligned}$$

Mismatch error       $r_n \frac{\lambda _n}{\lambda _{n+1} } \log \lambda _n \ll r_{n+1}\iff \lambda _n^{(b-1)(\beta -1)}\log \lambda _n\ll 1.$

Transport error     $\lambda _n^{1-\alpha }\sqrt{\frac{r_n}{\lambda _{n+1}}}\ll r_{n+1}\iff \lambda _n^{1-\alpha -\frac{1}{2} \beta -\frac{1}{2} b+b\beta }\ll 1.$

Dissipation error $\lambda _{n+1}^{\gamma -1}\sqrt{\frac{r_n}{\lambda _{n+1}}}\ll r_{n+1}\iff \lambda _{n+1}^{\gamma -\frac{3}{2}+\beta -\frac{\beta }{2b} }\ll 1.$

$C^\alpha$-regularity       $\lambda _{n+1}^\alpha \sqrt{\frac{r_n}{\lambda _{n+1}}}\ll 1 \iff \lambda _{n+1}^{\alpha -\frac{1}{2}-\frac{1}{2b} \beta }\ll 1.$

Now one can take $\alpha \approx \frac{1}{2} +\frac{\beta }{2b}$ to do a limiting computation. From the transport error we obtain (the limiting condition)

$$\begin{aligned} 1-\alpha -\frac{1}{2} \beta -\frac{1}{2} b +b \beta =\frac{1-b}{2b} (b-\beta (2b+1)) \Rightarrow \beta <\frac{1}{3}. \end{aligned}$$

From the dissipation error we obtain $\frac{\beta }{2} <\frac{3}{2} -\gamma$.