# 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\$ Answer

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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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