Displaying items by tag: Primitive equations

Quyuan Lin, Changhui Tan

Abstract

We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range \(1<\alpha\leq2\). Together with the known ill-posedness results for \(0\leq\alpha<1\), this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework.

The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the \(L^\infty\) norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.

	This work is supported by NSF grants DMS #2238219
	This work is supported by a USC VPR ASPIRE grant

 Changhui Tan, Zhuan Ye

Abstract

In this paper, we investigate the two-dimensional incompressible primitive equations with fractional horizontal dissipation. Specifically, we establish global well-posedness of strong solutions for arbitrarily large initial data when the dissipation exponent satisfies \(\alpha\geq\alpha_{0}\approx1.1108\). In addition, we prove global well-posedness of strong solutions for small initial data when \(\alpha \in [1, \alpha_0)\). Notably, the smallness assumption is imposed only on the \(L^\infty\) norm of the initial vorticity.

	This work is supported by NSF grants DMS #2238219
	This work is supported by a USC VPR ASPIRE grant

 Elie Abdo, Quyuan Lin, and Changhui Tan

Abstract

The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed.

In this paper, we study the two-dimensional incompressible PE with fractional horizontal dissipation. We identify a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent \(\alpha=1\). In the critical regime, this dichotomy exhibits a new phenomenon: the transition depends delicately on the balance between the size of the initial data and the viscosity coefficient. Our results precisely quantify the horizontal dissipation required to transition from inviscid instability to viscous regularity. We also establish a global well-posedness theory to the fractional PE, with sufficient dissipation \(\alpha\geq\frac65\).

	This work is supported by NSF grants DMS #2108264 and DMS #2238219
	This work is supported by a USC VPR ASPIRE grant