Stable Singularity Formation for the Inviscid Primitive Equations

Abstract

The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.

Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations

“…For the inviscid case, the ill-posedness can be overcomed by assuming either some special structures (local Rayleigh condition) on the initial data in 2D, or real analyticity in all directions for general initial data in both 2D and 3D [4,5,22,26,33,34,40]. While the strong solutions to the PEs with the horizontal viscosity ν h > 0 exist globally in time, it has been shown that smooth solutions to the inviscid PEs can develop singularity in finite time [7,13,30,48]. Whether the smooth solutions to the PEs with only vertical viscosity exist globally or blow up in finite time still remains open.…”

“…(ii) Unlike the stochastic PEs with ν h > 0 where the global existence of solutions can be proven, we can only show the local existence of martingale solutions. The main reason is that, in the deterministic case, the solutions to the PEs with ν h = 0 are unknown to exist globally when ν z > 0, and have been shown to form singularity in finite time when ν z = 0 [7,13,30,48]. (iii) Compared to the viscosity which can give "strong dissipation" by providing one gain in the derivative, the analytic framework only gives "weak dissipation" by providing one-half gain in the derivative.…”

Pathwise Solutions for Stochastic Hydrostatic Euler Equations and Hydrostatic Navier-Stokes Equations Under the Local Rayleigh Condition

“…Such ill-posedness can be overcome in the following two situations: 1) In the 2D case, the local well-posedness can be obtained by assuming the initial data satisfying the local Rayleigh condition (1.3) [7,45]; 2) By assuming real analyticity in all directions for general initial data in both 2D and 3D, [24,40] established the local well-posedness in the space of analytic functions with the radius of analyticity shrinking in time. Unlike the case with horizontal viscosity where the strong solutions exist globally in time, the smooth solutions to the inviscid PEs have been shown to form singularity in finite time [10,16,34,53].…”

Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations

“…(ii) Unlike the stochastic PEs with ν h > 0 where the global existence of solutions can be proven, we can only show the local existence of martingale solutions. The main reason is that, in the deterministic case, the solutions to the PEs with ν h = 0 are unknown to exist globally when ν z > 0, and have been shown to form singularity in finite time when ν z = 0 [7,13,29,46]. (iii) Compared to the viscosity which can give "strong dissipation" by providing one gain in the derivative, the analytic framework only gives "weak dissipation" by providing one-half gain in the derivative.…”

“…For the inviscid case, the ill-posedness can be overcomed by assuming either some special structures (local Rayleigh condition) on the initial data or real analyticity in all directions for general initial data [4,5,21,25,32,33,39]. While the strong solutions to the PEs with the horizontal viscosity ν h > 0 exist globally in time, it has been shown that smooth solutions to the inviscid PEs can develop singularity in finite time [7,13,29,46]. Whether the smooth solutions to the PEs with only vertical viscosity exist globally or blow up in finite time still remains open.…”