on Riemann Hypothesis and the proving


254A, Notes 1: Local well-posedness of the Navier-Stokes equations

What's new

We now begin the rigorous theory of the incompressible Navier-Stokes equations

$latex displaystyle partial_t u + (u cdot nabla) u = nu Delta u – nabla p (1)&fg=000000$

$latex displaystyle nabla cdot u = 0,&fg=000000$

where $latex {nu>0}&fg=000000$ is a given constant (the kinematic viscosity, or viscosity for short), $latex {u: I times {bf R}^d rightarrow {bf R}^d}&fg=000000$ is an unknown vector field (the velocity field), and $latex {p: I times {bf R}^d rightarrow {bf R}}&fg=000000$ is an unknown scalar field (the pressure field). Here $latex {I}&fg=000000$ is a time interval, usually of the form $latex {[0,T]}&fg=000000$ or $latex {[0,T)}&fg=000000$. We will either be interested in spatially decaying situations, in which $latex {u(t,x)}&fg=000000$ decays to zero as $latex {x rightarrow infty}&fg=000000$, or $latex {{bf Z}^d}&fg=000000$-periodic (or periodic for short) settings, in which one has $latex {u(t, x+n) = u(t,x)}&fg=000000$ for all $latex {n in {bf Z}^d}&fg=000000$. (One…

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