[Paper Review] Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data

Chongsheng Cao, Jinkai Li+1 more|arXiv (Cornell University)|Jul 21, 2016

Advanced Mathematical Physics Problems29 references5 citations

TL;DR

This paper establishes local and global well-posedness of strong solutions to the 3D primitive equations with only horizontal dissipation, for initial data in $H^1$ and $H^1 \cap L^\infty$ with $\partial_z v_0 \in L^m$ for $m \in (2,\infty)$. It achieves this using anisotropic Sobolev inequalities and a logarithmic Gronwall inequality, improving prior results that required $H^2$ initial data.

ABSTRACT

In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data $(v_0, T_0)\in H^1$, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0, T_0)\in H^1\cap L^\infty$, such that $\partial_zv_0\in L^m$, for some $m\in(2,\infty)$, by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where the initial data $(v_0, T_0)$ was assumed to have $H^2$ regularity.

Motivation & Objective

• To establish local well-posedness of strong solutions to the 3D primitive equations with only horizontal viscosity and diffusivity for initial data in $H^1$.
• To extend global well-posedness to initial data in $H^1 \cap L^\infty$ with $\partial_z v_0 \in L^m$, $m \in (2,\infty)$, improving prior results requiring $H^2$ regularity.
• To develop and apply a logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality to control the solution's growth.
• To validate the physical relevance of the model by showing global existence without vertical dissipation, consistent with oceanic and atmospheric dynamics.

Proposed method

• Utilizes local-in-space energy estimates to prove local well-posedness for $H^1$ initial data.
• Employs anisotropic Sobolev inequalities to control the vertical derivative of velocity in terms of horizontal derivatives.
• Applies a logarithmic type Gronwall inequality to control the growth of high-order norms over time.
• Uses the structure of the primitive equations with only horizontal dissipation to derive energy estimates that avoid reliance on vertical viscosity.
• Implements a change of variables and weighted integration to handle the anisotropic nature of the dissipation.
• Relies on the Newtonian potential representation and integration by parts to derive pointwise bounds on the solution.

Experimental results

Research questions

• RQ1 Can strong solutions to the 3D primitive equations with only horizontal dissipation be shown to exist locally for initial data in $H^1$?
• RQ2 What regularity conditions on initial data are sufficient to ensure global existence of strong solutions when vertical dissipation is absent?
• RQ3 How can anisotropic Sobolev inequalities be used to control the solution's behavior in the absence of vertical viscosity?
• RQ4 Can a logarithmic Gronwall inequality be applied effectively to control the growth of $H^1$-type norms in this system?
• RQ5 Is it possible to reduce the required initial regularity from $H^2$ to $H^1 \cap L^\infty$ with mild vertical derivative integrability?

Key findings

• Local well-posedness of strong solutions is established for any initial data $(v_0, T_0) \in H^1$ using local-in-space energy estimates.
• Global well-posedness is proven for initial data $(v_0, T_0) \in H^1 \cap L^\infty$ with $\partial_z v_0 \in L^m$, $m \in (2,\infty)$, via anisotropic Sobolev and logarithmic Gronwall inequalities.
• The result improves upon prior work by reducing the required initial regularity from $H^2$ to $H^1 \cap L^\infty$ with mild vertical derivative control.
• The logarithmic type Gronwall inequality effectively controls the growth of the solution's $H^1$-norm over time.
• The method confirms the physical consistency of the model by showing global existence without vertical dissipation.
• The analysis demonstrates the effectiveness of anisotropic estimates in handling systems with strong horizontal and weak vertical dissipation.