

@mastersthesis {6759,
	title = {Well-posedness of nonlinear evolution equations},
	year = {1996},
	note = {9704179},
	school = {University of California, Santa Barbara},
	type = {phdPh.{D}.},
	address = {Ann Arbor},
	abstract = {This thesis is concerned with local properties of three nonlinear evolution systems: the generalized Burgers{\textquoteright} equation, Benney{\textquoteright}s interaction system for long and short water waves, and a coupled Schrodinger-Korteweg de Vries system. For each system well-posedness of the Cauchy problem is shown. For the generalized Burgers{\textquoteright} equation,\$\${\textbackslash}left{\textbackslash}\{\vphantom{\}} {\textbackslash}eqalign\{\&{\textbackslash}partial{\textbackslash}sb\{t\}u-{\textbackslash}partial{\textbackslash}sbsp\{x\}\{2\}u+{\textbackslash}partial{\textbackslash}sb\{x\}(u{\textbackslash}sp\{k + 1\}) = 0,{\textbackslash}quad t, x{\textbackslash}in(0,T){\textbackslash}times{\textbackslash}IR,{\textbackslash}cr \&u(x,0)=u{\textbackslash}sb0(x),\}{\textbackslash}cr\$\$well-posedness in sought in the general Sobolev spaces, \${\textbackslash}dot L{\textbackslash}sp\{p,s\}({\textbackslash}IR)\$ and \$L{\textbackslash}sp\{p,s\}({\textbackslash}IR)\$. It is shown that the equation is well-posed up to and including the scaling exponent, \$s=s{\textbackslash}sb\{p,k\} := 1/p-1/k.\$ Next, for Benney{\textquoteright}s system,\$\${\textbackslash}left{\textbackslash}\{\vphantom{\}} {\textbackslash}eqalign\{\&i{\textbackslash}partial{\textbackslash}sb\{t\}u+{\textbackslash}partial{\textbackslash}sbsp\{x\}\{2\}u = {\textbackslash}alpha{\textbackslash}vert u{\textbackslash}vert{\textbackslash}sp2u+{\textbackslash}beta nu,{\textbackslash}quad t, x{\textbackslash}in(0,T){\textbackslash}times{\textbackslash}IR,{\textbackslash}cr \&{\textbackslash}partial{\textbackslash}sb\{t\}n={\textbackslash}partial{\textbackslash}sb\{x\}({\textbackslash}vert u{\textbackslash}vert{\textbackslash}sp2){\textbackslash}cr \&u(x,0)=u{\textbackslash}sb0(x),{\textbackslash}cr \&n(x, 0) = n{\textbackslash}sb0(x),\}{\textbackslash}cr\$\$well-posedness is shown for \$(u{\textbackslash}sb0, n{\textbackslash}sb0){\textbackslash}in H{\textbackslash}sp\{s\}({\textbackslash}IR){\textbackslash}times L{\textbackslash}sp\{1/s\}({\textbackslash}IR)\$ for the case \${\textbackslash}alpha=0\$ and \$s{\textbackslash}in(0, 1/2).\$ For the case \$1/2{\textbackslash}le s{\textless}1\$ and \${\textbackslash}alpha{\textbackslash}not= 0\$ it is shown that given \${\textbackslash}varepsilon {\textgreater} 0,\$ and (\$u{\textbackslash}sb0, n{\textbackslash}sb0){\textbackslash}in H{\textbackslash}sp\{s\}({\textbackslash}IR){\textbackslash}times H{\textbackslash}sp\{s-(1/2)+{\textbackslash}varepsilon\}({\textbackslash}IR)\$ Benney{\textquoteright}s system is well-posed. For the case \$s=1/2\$ then \${\textbackslash}varepsilon=0\$ suffices. Furthermore, if \${\textbackslash}alpha=0\$ the solution is global. Lastly, an interaction system of the capillary-gravity wave is considered. It is shown that the Cauchy problem of a coupled Schrodinger-Korteweg de Vries system,\$\${\textbackslash}left{\textbackslash}\{\vphantom{\}} {\textbackslash}eqalign\{\&i{\textbackslash}partial{\textbackslash}sb\{t\}u+{\textbackslash}partial{\textbackslash}sbsp\{x\}\{2\}u={\textbackslash}alpha{\textbackslash}vert u{\textbackslash}vert{\textbackslash}sp2u+{\textbackslash}beta vu,{\textbackslash}quad t, x{\textbackslash}in(0,T){\textbackslash}times{\textbackslash}IR,{\textbackslash}cr \&{\textbackslash}partial{\textbackslash}sb\{t\}v+{\textbackslash}partial{\textbackslash}sbsp\{x\}\{3\}v+{\textbackslash}partial{\textbackslash}sb\{x\}v{\textbackslash}sp2={\textbackslash}gamma {\textbackslash}partial{\textbackslash}sb\{x\}({\textbackslash}vert u{\textbackslash}vert{\textbackslash}sp2),{\textbackslash}cr \&u(x,0)=u{\textbackslash}sb0(x),{\textbackslash}cr \&v(x,0)=v{\textbackslash}sb0(x),{\textbackslash}cr\}\$\$is locally well-posed for \$(u{\textbackslash}sb0, v{\textbackslash}sb0){\textbackslash}in L{\textbackslash}sp2({\textbackslash}IR){\textbackslash}times H{\textbackslash}sp\{-1/2\}({\textbackslash}IR).\$},
	keywords = {0405:Mathematics, Mathematics, Pure sciences},
	url = {http://search.proquest.com/docview/304287630?accountid=14270},
	author = {Bekiranov, Daniella}
}
