| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Global solutions to SQG equation

The Surface Quasi-Geostrophic (SQG) equation is a simple 2D model, whose dynamics still nead to be understood.
    1. Existence of global weak solutions

      Problem 1.1.

      [Andrea Nahmod] The modified SQG (mSQG) equation reads \begin{aligned} \partial_t\theta +u\cdot \nabla\theta &=0, \\ u &= \mathcal{R}^\perp|\nabla|^{-\varepsilon}\theta, \end{aligned} where $\theta : \mathbb{R}\times\mathbb{R}^2\to\mathbb{R}$, $\varepsilon>0$, and $\mathcal{R}$ is the Riesz transform. Is it possible to find global weak solutions to (mSQG) when $\varepsilon=0$, with respect to random initial data ?
        • Towards strong global solutions ?

          Problem 1.2.

          [Alex Ionescu] Is it possible to describe the formation of singularities in (mSQG) ?
            • About uniqueness

              Problem 1.3.

              [Gigliola Staffilani] Find a suitable notion of "probabilistic uniqueness" for solutions of SQG equations.

                  Cite this as: AimPL: Mathematical questions in wave turbulence theory, available at http://aimpl.org/waveturb.