Partial Differential Equations applied to Oceanography and Classical Fluid Mechanics

VasseurDMS-1209420 调查员地址的定性性质的解决方案的纳维-斯托克斯，欧拉，准地转，和其他重要的方程在流体动力学。 他研究气象学，海洋学和流体力学的非线性问题，使用最先进的数学方法分析和非线性偏微分方程理论。 他继续研究三维流体力学的规律性理论。 特别强调的是，与欧拉方程相比，Navier-Stokes方程的环流不守恒的后果。 分数阶扩散算子对于模拟被动或主动标量的湍流运动或滞后现象（具有记忆）至关重要。 他对这一系列方程进行了系统的研究。 研究人员专注于流体力学中涉及的数学模型。 Navier-Stokes方程和Euler方程是研究流体动力学的基本方程组。 了解解的行为在工程、气象学、医学和海洋学中具有根本的重要性。 当追踪大气现象的行为时，这些方程对于分析处理来说太复杂了，这对气候变化的研究至关重要。 准地转方程是一个简化的方程，从基本方程推导出来，它仍然能捕捉到大尺度流动的重要动力学。 调查员对海洋学中常用的这一重要方程式进行了系统研究。

VasseurDMS-1209420 The investigator addresses the qualitative properties of solutions to the Navier-Stokes, Euler, quasi-geostrophic, and other important equations in fluid dynamics. He studies nonlinear problems from meteorology, oceanography, and fluid mechanics, using state of the art mathematical methods from analysis and the theory of nonlinear partial differential equations. He continues his investigation of regularity theory in 3D fluid mechanics. A particular emphasis is the consequences of the non-conservation of circulation for the Navier-Stokes equation compared to the Euler equation. Fractional diffusion operators are crucial to model turbulent motion of passive or active scalars, or hysteresis phenomena (with memory). He develops a systematic study of this family of equations. The investigator focuses on mathematical models involved in fluid mechanics. The Navier-Stokes and Euler equations are fundamental systems to study the dynamics of fluids. The understanding of behavior of the solutions is of fundamental importance in engineering, meteorology, medicine, and oceanography. When tracking the behavior of atmospheric phenomena, crucial for the study of climate change, those equations prove too complex for analytical treatment. The quasi-geostrophic equation is a simplified equation, derived from the fundamental equations, that still captures important dynamics at large scales of the flow. The investigator undertakes a systematic study of this important equation, which is commonly used in oceanography.