|1 1||2 4||3||4 2||5|
|6||7 2||8 4||9||10 2||11 1||12|
|13||14||15 1||16||17 3||18||19|
|20||21||22 1||23 1||24||25||26|
|27||28 1||29 2||30 1||31|
|10||11||12 2||13||14 1||15||16|
|17||18||19 2||20||21 2||22 1||23|
The Boussinesq abcd system was originally derived by Bona, Chen and Saut [J. Nonlinear. Sci. (2002)] as a rst order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime. Among many particular regimes, the Hamiltonian generic regime is characterized by the setting b = d > 0 and a; c < 0. It is known that the system in this regime is globally well-posed for small data in the energy space H1 H1 by Bona, Chen and Saut [Nonlinearity (2004)]. In this talk, we are going to discuss about the decay of small solutions to abcd system in three directions: First, for a weakly dispersive abcd systems (characterized only in terms of parameters a; b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone jxj jtj. Second, for every ray x = vt, jvj < 1 inside the light cone, small solutions to suciently dispersive system (smallness and dispersion are characterized by v) decay to zero, in proper subset along the ray. Last, small solutions decay to zero in exterior regions jxj jtj under suitable conditions of parameters (a; b; c). All results rule out, among other things, the existence of zero or nonzero speed or super-luminical small solitary waves in each regime where decay is present.
This is joint work with Claudio Munoz.
This is joint work with Sławomir Kołodziej. We show that the complex m-Hessian
operator of a Holder continuous m-subharmonic function is well dominated by the corresponding capacity. As consequence we obtain the Holder continuous subsolution theorem for the complex m-Hessian equation.