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## fa.purposeful evaluation – Example of BV vector bailiwick $c$ with out bounded divergence such that $u$ is bounded the place $u_t + div(cu) = 0$

the situation of bounded divergence in wanted! for the reason that contrary signature between $u_t$ and $div(cu)$ suggest, underneath its border situation, there should exists a course of that its answer rot as a hurry of exponent, clear by its border worth. since divergence in an equation delineate its root’s strengthen. if it has constructive root, attribute $lambda$ and its relation sever with time will learn its hurry of rot.

for instance, write $divF=bigtriangledowncdot{F}=frac{partial{F_x}}{partial{x}}+frac{partial{F_y}}{partial{y}}$

then we outline a separated variables of $u(x,y,t)=h(t)phi(x,y)$, the place $frac{dh}{dt}=-lambda{kh},frac{partial^2{phi}}{partial{x}^2}+frac{partial^2{phi}}{partial{y}^2}=-lambda{phi}$.

due to this fact, its answer $u$ is bounded above.

nevertheless, if its answer will not be situated in a sole route, in any other case $u$ can breathe a traverse worth. we should always employ finite dissimilarity system to approximate, however the native situation in BV vector require to choose an epsilon ball inside $bigtriangledown_{x}b(t,x)y$, as a substitute of $W^{1,1}$, to conform its equilibrium, not allowable to occupy $Delta{x}=Delta{y}$, thus not allowable to assemble an analytical approximation with $u^{(m)}_{j+1}-2u^{(m)}_{j}+u^{(m)}_{j-1}$, underneath the route of former and backward.

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