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