Large Deviations For The 2-D Stochastic Derivative Ginzburg–Landau With Multiplicative Noise

The Ginzburg-Landau equation is an important model of superconductivity and is considered to be a fundamental equation in modern physics.The deterministic Ginzburg-Landau equation,i.e.without the derivative term and the random term,describes the various pattern formations and the onset of instabilities in nonequilibrium fluid dynamical systems and the theory of phase transitions and superconductivity.In the present paper,we introduce a multiplicative noise with the form ?(u)dW(t)/ dt into the deterministic derivative Ginzburg-Landau equation and consider its stochastic version.The noise models small irregular fluctuations generated by microscopic effects in the flux of molecular collisions and thus the stochastic model may be more realistic.In this paper,the local existence of mild solutions to the two-dimensional stochastic derivative Ginzburg-Landau equation is obtained by using the Banach contraction principle.Then,the energy estimate shows that the solution is also global in time.Since this paper considers the two-dimensional case,some Banach spaces are introduced and the stochastic estimates in these Banach spaces are necessary.Secondly,a Freidlin-Wentzell type large deviation principle is established for the two-dimensional stochastic derivative Ginzburg-Landau equation perturbed by a small white noise.Since the underlying space is Polish,the large deviation principle is equivalent to the Laplace principle.Therefore,we use the weak convergence method to establish the large deviation principle by proving the Laplace principle.Finally,the large deviation principle of the two-dimensional stochastic derivative Ginzburg-Landau equation for short time is also considered.