Regularity of 2D Surface Quai-Geostrophic (SQG) Equations

Abstract

In this research, we delve into three distinct topics within the realm of nonlinear fluid dynamics, namely the generalized Korteweg-de Vries (KdV)-type equation, the regularity of solutions in the $2$D Surface Quasi-Geostrophic (SQG) equation, and the behavior of water waves under indefinite boundary constraints.

Firstly, we undertake an analytical and numerical examination of the following generalized KdV-type equation ut+aux+2buux+cuxxx- duxx=0, u(x,0)=u0(x) where a, b, c, d are real parameters.

Our study involves allowing the coefficients a, b, c , and d to approach zero in the limiting sense, while contrasting the outcomes with the scenario in which each coefficient is precisely zero. By analyzing this nonlinear partial differential equation in one dimension, we trace the impact of the nonlinear term on the solution. Furthermore, we extend our findings to a two-dimensional equation with structures comparable to those in the 2D SQG equation.

Secondly, we focus on the regularity of solutions in the following 2D SQG equation

where κ ≥ 0 and α > 0 are parameters,

conducting a thorough analysis that addresses a notable gap in analytical and numerical research. The SQG equation exhibits numerous characteristics similar to the 3D Euler equation and the Navier-Stokes equation, with the regularity of the latter being recognized as one of the Clay Institute of Mathematics' millennium problems. To bridge this gap, we concentrate on various aspects of the SQG equation, exploring both inviscid and dissipative instances. In the dissipative case, we categorize the instances as subcritical, critical, and supercritical. Analytical solutions have recently been derived for the subcritical and critical scenarios, while the question of regularity in the supercritical case remains unresolved. Our research focuses on numerical calculations of the inviscid and supercritical SQG equations, with particular attention to the proximity of level curves, the L2 norm, and the expansion of the quantity. We meticulously examine the nature of the solution, particularly in the region where α =.

Finally, we turn our attention to the study of the following water waves

where u is the velocity, P is the pressure, and g is the acceleration due to gravity,

which are typically modeled using Euler equations with unit density. We address an outstanding open problem concerning the existence of closed orbits for water waves under indefinite boundary constraints. Our investigation begins with a discussion of advancements in water wave structure under finite bottom conditions. We then shift our focus to the behavior of water waves at the kinematic barrier of infinite depth. By employing the Crandall-Rabinowitz theorem to construct water wave profiles for scenarios with zero and