Abstract
We consider the Cauchy problem of the nonlinear Schrödinger equation with magnetic effect, and prove global existence of smooth solutions and decay estimates for suitably small initial data. The key step in our analysis is to exploit the null structures for the phases, which allow us to close our argument in the framework of space-time resonance method.
Highlights
We are concerned with the global existence of smooth solutions for the Cauchy problem of a set equations arising from plasma physics
E : R+ × R3 → C3 is the slowly varying amplitude of the high-frequency electric field (E denotes the complex conjugate of E), and the function B : R+ × R3 → R3 is the self-generated magnetic field. γ is the speed of electron, and the notation × means the cross product for R3 or C3 valued vectors
System (1) is a simplified model in plasma physics. It describes the nonlinear interaction between plasma-wave and particles [22], especially when the phase speed of plasma wave is much less than the speed of ions so that the fluctuation of the density satisfies a stationary equation
Summary
We are concerned with the global existence of smooth solutions for the Cauchy problem of a set equations arising from plasma physics. In the estimates for the energy norm, since the regularity of E and M is not at the same level and the nonlinear term contains two order derivatives, the usual energy method will introduce one order loss of derivative. To overcome this difficulty, we use the idea of [21] to exploit the following positive properties.
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