Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation

Abstract For any initial datum $$\theta _0\in L^{\frac{4}{3}}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msubsup> </mml:mrow> </mml:math> it is proven that the existence of a global-in-time weak solution $$\theta \in L^\infty _t L^{\frac{4}{3}}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msubsup> </mml:mrow> </mml:math> to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $$\dot{H}^{-\frac{1}{2}}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msubsup> </mml:math> norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the $$L^{\frac{4}{3}}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msubsup> </mml:math> norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.