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

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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# Probability distribution on coordinates

using axis to represent variables. The coordinate system can be multidimentional.

Representing the probability of a certain event as a distribution in space. The distribution will turn out to be not clustered or very discrete if the event is not based on a summation-like condition of  the variables.

# Sibelius- Symphony No.5 in E-Flat Major, Op.82 III. Allegro Molto

It was the third chapter that attracted my attention.

How does the numbering system of a music piece work?

elements of the music: paragraphs; main song; ….

Sibelius and background of this music.

# Hello World!

The blog is designated to keep track of interesting discoveries and possibly valuable ideas.

Some of these are extensions from my class or research material, and some would be purely hobby.