Local Well-Posedness of a Coupled Camassa-Holm System in Critical Spaces

2015 ◽  
Vol 34 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Xingxing Liu
Keyword(s):  
Well Posedness ◽  
Critical Spaces

scholarly journals Random initial conditions for semi-linear PDEs

2019 ◽  
Vol 150 (3) ◽  
pp. 1533-1565
Author(s):  
Dirk Blömker ◽  
Giuseppe Cannizzaro ◽  
Marco Romito
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Stochastic Pdes ◽  
Well Posedness ◽  
Critical Spaces ◽  
Random Initial Conditions ◽  
Linear Pdes ◽  
Point Argument

AbstractWe analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

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