A Summary Of Carleman Inequality Of Parabolic Equations And Its Applications

In this summary we introduce the Carleman inequality of parabolic equations and its applications in the controllability, strong unique continuation etc.Since the Carleman inequality plays a critical role in the controllability, unique continuation and the inverse problem of the evolutionary equations(such as the wave equation, the heat equation, Schrodinger equation, Navier-Stokes equations), in the recent decade, it has already attracted the attention of many mathematicians. For the linear and semilinear evolutionary equations, establishing the Carleman inequality and hence obtaining the observability inequality has been a classical fashion to treat the controllability problem under the framework of the duality principle and HUM(Hilbert Unique Method) method. It is also one of the most powerful technique in investigating the controllability of the evolutionary equations.This paper main introduce the proof of Carleman inequalities and its application in controllability in recent decades. We give the introduction of the Carleman inequalities and its applications on the linear parabolic equation, the semilinear parabolic equations, the parabolic systems etc.