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MA4G5 Content

Content:
Topics include:

  • The equations (brief derivation and key properties) 1
  • The vorticity formulation and Biot–Savart law
  • Local-in-time existence and uniqueness results in Rn, n = 2, 3, via energy estimates
  • An alternative approach to local well-posedness for Euler, using particle trajectory methods
  • Global-in-time existence results in 2D and comparisons to 3D
  • Criteria for blowup of solutions e.g. the celebrated Beale-Kato-Majda theorem
  • An introduction to weak solutions of the Navier–Stokes equations
  • A global existence result for weak solutions of the Navier–Stokes equations (time permitting)
  • Other selected topics, according to student interest (time permitting)

Aims:
This course aims to give an introduction to the rigorous analytical theory of the PDEs of fluid mechanics. In particular we will focus on the incom-pressible Euler and Navier-Stokes equations in R2 and R3, which are widely used models for inviscid and viscous flow, respectively. The questions of global existence and uniqueness of solutions to these systems form the basis for a great deal of current research. In this course we will study a few of the fundamental results in this field, which will give students a chance to apply knowledge from Functional Analysis and PDE modules to these highly-relevant non-linear systems.

Objectives:
By the end of the module, students will:

  • Be familiar with the Euler and Navier–Stokes and the physical meaning of the terms therein, for classical and vorticity-stream formulations.
  • Have explored, in these particular cases, some of the typical issues arising in the study of PDEs (local vs global existence, uniqueness, blowup criteria, 2D vs 3D behaviour etc. )
  • Have learnt two approaches to proving local existence and uniqueness restults: via an energy methods (featuring Sobolev estimates), and a particle-trajectory method (using H¨older spaces).
  • Have seen the definition of a weak solution of the Navier–Stokes equa-tions and a discussion of further well-known existence results (at least in summary).

Books:

  • Primary text: A.J. Majda and A.L. Bertozzi. Vorticity and incom¬pressible flow. CUP, Cambridge, 2002.
  • A.J. Chorin and J.E. Marsden. A mathematical introduction to fluid mechanics, volume 4 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition, 1993.
  • J.C. Robinson, J.L. Rodrigo, and W. Sadowski. The three–dimensional Navier–Stokes equations. Classical Theory. Cambridge University Press, Cambridge, 2016.
  • P. Constantin and C. Foias. Navier-Stokes Equations. The University of Chicago Press, Chicago, 1988.