Fluid mechanics

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• ConceptShallow water equations

  The shallow water equations describe a layer of fluid whose depth is small compared with the horizontal length scale of its motion. Their unknowns are the depth of the layer and the horizontal velocity, both depending on horizontal position and time. They form a system of hyperbolic conservation laws for mass and momentum. They model tides, tsunamis, floods and large-scale flows in the atmosphere and the ocean.

• ConceptKorteweg–De Vries equation

  The Korteweg–De Vries equation, or KdV equation, is the nonlinear partial differential equation $u_t + u_{xxx} + 6 u u_x = 0$ for a real function $u(x,t)$. It models long waves of small amplitude in shallow water. It has solitary wave solutions that keep their shape, and it is the first equation that was solved by the inverse scattering transform. It has infinitely many conserved quantities and is a central example of a completely integrable system.

• ConceptKadomtsev–Petviashvili equation

  The Kadomtsev–Petviashvili equation, or KP equation, is a nonlinear wave equation in two space variables $x$ and $y$. It extends the Korteweg–De Vries equation to waves that move mainly in the $x$ direction but vary slowly in the $y$ direction. It comes in two versions, KP-I and KP-II, which differ by the sign of the $u_{yy}$ term. Like the KdV equation it is completely integrable.

• ConceptNavier–Stokes equations

  The Navier–Stokes equations describe the motion of a viscous fluid, such as water or air. They are Newton's second law for each small parcel of fluid, with forces from pressure, internal friction and external sources. In the incompressible case they form a nonlinear system for the velocity field and the pressure. Whether smooth solutions in three dimensions stay smooth for all time is one of the Millennium Prize Problems.

• ConjectureNavier–Stokes existence and smoothness

  Let $\nu > 0$. Let $u_0 \colon \R^3 \to \R^3$ be a smooth vector field with divergence $\nabla \cdot u_0 = 0$. Assume that for every multi-index $\alpha$ and every $K > 0$ there is a constant $C_{\alpha K}$ with $\abs{\partial^\alpha u_0(x)} \le C_{\alpha K} (1 + \abs{x})^{-K}$ for all $x \in \R^3$. Then there exist smooth functions $u \colon \R^3 \times [0, \infty) \to \R^3$ and $p \colon \R^3 \times [0, \infty) \to \R$ such that:

• ConceptStokes flow

  Stokes flow is the flow of a viscous fluid in which viscous forces are much larger than inertial forces. It is governed by the Stokes equations, the linear system obtained from the Navier–Stokes equations by dropping the nonlinear term and the time derivative. Because the equations are linear, solutions can be added and many problems can be solved exactly. Stokes flow models very small organisms, slow flows of very viscous fluids and flows in thin films.