In the second part of this thesis, we consider the aggregation equation ut + 1.asqbl0;aparl0;1Kaparr0;*uaparr0;uasqbr0;=0 with nonnegative initial data in L1Rn abigcap;L infinityRn for n ay 2. We assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuation of solutions, with the bulk of the work devoted to proving existence. Because our PDE is a transport equation, existence for initial data in L1Rn abigcap;L infinityRn cannot be established using energy methods. Instead, we adapt ideas from the solution of the 2-D vorticity equation for the same class of initial data. Finite time blow-up (in the Linfinity norm) of solutions to our problem is proved when the kernel has precisely a Lipschitz point at the origin.We first estimate //; integrating by parts, applying Younga#39;s inequality for convolutions and the Cauchy-Schwartz inequality, we arrive at \n\ alt; I|w|Il2(Raquot;) IIuiIUoAd(kArII^2-Ra#39;IIli(rAr)- We may also ... In two dimensions, we can re-write / as / ( Qx4agt;x +anbsp;...
|Title||:||A Level-set Method for Solving Elliptic Eigenvalue Problems on Hypersurfaces, And, Finite-time Blow-up of L(infinity) Weak Solutions of an Aggregation Equation|
|Publisher||:||ProQuest - 2008|