Zakharov system on R^2: Difference between revisions
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Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1/2,-1). | Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1/2,-1). | ||
LWP for (s0,s1) = (1/2,0) [GiTsVl1997] | LWP for (s0,s1) = (1/2,0) [[GiTsVl1997]] | ||
For (s0,s1) = (1,0) this was proven in [BoCo1996], [Co1997]. | For (s0,s1) = (1,0) this was proven in [[BoCo1996]], [[Co1997]]. | ||
GWP for small (1,0) data [BoCo1996]; the smallness is needed to control the nonquadratic portion of the energy. | GWP for small (1,0) data [[BoCo1996]]; the smallness is needed to control the nonquadratic portion of the energy. | ||
As long as the H1 norm remains bounded (which is automatic for small data), the higher H^s norms of u grow by at most |t|(s-1)+ [CoSt-p] | As long as the H1 norm remains bounded (which is automatic for small data), the higher H^s norms of u grow by at most |t|(s-1)+ [[CoSt-p]] | ||
Explicit blowup solutions have been constructed with a blowup rate of t-1 in H1 norm [GgMe1994], [GgMe1994b]. This is optimal in the sense that no slower blowup rate is possible [Me1996b] | Explicit blowup solutions have been constructed with a blowup rate of t-1 in H1 norm [[GgMe1994]], [[GgMe1994b]]. This is optimal in the sense that no slower blowup rate is possible [[Me1996b]] | ||
[[Category:Equations]] | [[Category:Equations]] | ||
Latest revision as of 00:03, 3 February 2007
Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1/2,-1).
LWP for (s0,s1) = (1/2,0) GiTsVl1997
For (s0,s1) = (1,0) this was proven in BoCo1996, Co1997.
GWP for small (1,0) data BoCo1996; the smallness is needed to control the nonquadratic portion of the energy.
As long as the H1 norm remains bounded (which is automatic for small data), the higher H^s norms of u grow by at most |t|(s-1)+ CoSt-p
Explicit blowup solutions have been constructed with a blowup rate of t-1 in H1 norm GgMe1994, GgMe1994b. This is optimal in the sense that no slower blowup rate is possible Me1996b
