<p>For any initial datum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta _0\in L^{\frac{4}{3}}_x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>∈</mo> <msubsup> <mi>L</mi> <mi>x</mi> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </msubsup> </mrow> </math></EquationSource> </InlineEquation> it is proven that the existence of a global-in-time weak solution <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta \in L^\infty _t L^{\frac{4}{3}}_x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mi>t</mi> <mi>∞</mi> </msubsup> <msubsup> <mi>L</mi> <mi>x</mi> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </msubsup> </mrow> </math></EquationSource> </InlineEquation> to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\dot{H}^{-\frac{1}{2}}_x$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mi>x</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{\frac{4}{3}}_x$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>x</mi> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </msubsup> </math></EquationSource> </InlineEquation> 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.</p>

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

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2026年8月3日

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Luigi De Rosa,
Mickaël Latocca,
Jaemin Park

摘要

For any initial datum $\theta _0\in L^{\frac{4}{3}}_x$ $θ_{0} \in L_{x}^{\frac{4}{3}}$ it is proven that the existence of a global-in-time weak solution $\theta \in L^\infty _t L^{\frac{4}{3}}_x$ $θ \in L_{t}^{\infty} L_{x}^{\frac{4}{3}}$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $\dot{H}^{-\frac{1}{2}}_x$ ${\dot{H}}_{x}^{- \frac{1}{2}}$ 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 $L^{\frac{4}{3}}_x$ $L_{x}^{\frac{4}{3}}$ 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.