Uniqueness Problem For Two Special Types Of Equations

The theory of uniqueness in the study of partial differential equations is an important issue. This paper mainly study the uniqueness issue of two types of equations. The first part get the unique continuation about the solution of Naiver-Stokes equations in Rn x [0, L]. And the second part prove uniqueness of the coefficient under the DN mapping in the bounded region Ω for the Schrodinger equations transformed by the Maxwell equation.The mathematical theory of Navier-Stokes equations has been a hot issue in mathemat-ics and physics. To simplify the equation, we particularly introduce the curl of the solution for the Navier-Stokes equations p. The original equation is reduced to the equation of p by the variable transformation. By selecting the appropriate weighting function, we want to get the estimate for the unique continuation with the weighted function of p. In order to end up the the estimated with the weighted function, we apply the inverse uniqueness to overcome the estimate difficulties caused by the region and get the estimation of p. Combined with the lemma has been known, we finally get the unique continuation about the solution u of Naiver-Stokes equations in Rn x [0, L].Then we study uniqueness of the coefficient under the DN mapping in a bounded region for the Schrodinger, namely the inverse boundary value problem which originally comes from the electrical impedance imaging technology. The problem can be reduced to study the inverse boundary problem associated to Schrodinger equations, and the selection of boundary size has been the focus of this issue. In this part, we transform the second-order equations into the a fourth-order equation. By using the Carleman estimate for bounded potential of fourth-order Schrodinger equation and the construction complex geometrical optics solutions, we get an integral identity related the potential q. By choosing an appropriate set of real analytic oscillating function and applying the results of analytic microlocal analysis, we obtain the uniqueness of its coefficients.