On energy conservation for the hydrostatic Euler equations: an Onsager conjecture

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jats:titleAbstract</jats:title>jats:pOnsager’s conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager’s conjecture in the context of the jats:italichydrostatic</jats:italic> Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In this case the relevant conserved quantity is the horizontal kinetic energy. We first consider the standard notion of weak solution which is commonly used in the literature. We show that if the horizontal velocity (jats:italicu</jats:italic>, jats:italicv</jats:italic>) is sufficiently regular then the horizontal kinetic energy is conserved. Interestingly, the spatial Hölder regularity exponent which is sufficient for energy conservation in the context of the hydrostatic Euler equations is jats:inline-formulajats:alternativesjats:tex-math$$\frac{1}{2}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mfrac mml:mn1</mml:mn> mml:mn2</mml:mn> </mml:mfrac> </mml:math></jats:alternatives></jats:inline-formula> and hence larger than the corresponding regularity exponent for the Euler equations (which is jats:inline-formulajats:alternativesjats:tex-math$$\frac{1}{3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mfrac mml:mn1</mml:mn> mml:mn3</mml:mn> </mml:mfrac> </mml:math></jats:alternatives></jats:inline-formula>). This is due to the anisotropic regularity of the velocity field: Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity jats:italicw</jats:italic> is one degree spatially less regular with respect to the horizontal variables, compared to the horizontal velocity (jats:italicu</jats:italic>, jats:italicv</jats:italic>). Since the standard notion of weak solution is not able to deal with this anisotropy properly, we introduce two new notions of weak solutions for which the vertical part of the nonlinearity is interpreted as a paraproduct. We finally prove several sufficient conditions for such weak solutions to conserve energy. </jats:p>


Funder: Cambridge Trust; doi: http://dx.doi.org/10.13039/501100003343

Funder: Cantab Capital Institute for Mathematics of Information

Funder: Hendrik Muller Fund

Funder: Alexander von Humboldt-Stiftung; doi: http://dx.doi.org/10.13039/100005156

4904 Pure Mathematics, 49 Mathematical Sciences
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Calculus of Variations and Partial Differential Equations
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Springer Science and Business Media LLC
Engineering and Physical Sciences Research Council (EP/K032208/1)