Partial Differential Equations applied to Oceanography and Classical Fluid Mechanics

研究者解决了流体动力学中Navier-Stokes， Euler，准地转方程和其他重要方程的解的定性性质。他研究气象学、海洋学和流体力学中的非线性问题，利用最先进的数学方法和非线性偏微分方程理论进行分析。他继续研究三维流体力学中的规律性理论。特别强调的是与欧拉方程相比，纳维-斯托克斯方程的循环不守恒的后果。分数扩散算子对于模拟被动或主动标量的湍流运动或迟滞现象（具有记忆）至关重要。他对这组方程进行了系统的研究。研究者专注于流体力学中的数学模型。纳维-斯托克斯方程和欧拉方程是研究流体动力学的基本系统。对溶液行为的理解在工程、气象学、医学和海洋学中是至关重要的。在追踪对气候变化研究至关重要的大气现象的行为时，这些方程被证明过于复杂，无法进行分析处理。准地转方程是由基本方程推导而来的简化方程，它仍然能捕捉到大尺度流动的重要动力学。研究者对这个重要的方程式进行了系统的研究，它通常用于海洋学。

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.