Unique Continuation Of Navier-Stokes Equations

Navier-Stokes equations is a kind of equations which describe the conservation of fluidmotion among hydromechanics, it can explain kinds of physical phenomena in our life,such as the airflow around the wing of the aircraft, the design of aircraft, the flow of liquidin pipeline. The mathematical theory research about solution of Navier-Stokes equations hasbeen received extensive attention. Many mathematics and physics researchers carried on thethorough research to it and made a lot of research achievements, but like these seeminglysimple physical problems, we didn’t get very good explanation in math. At present wemainly get the existence of weak solutions and uniqueness of strong solutions. However,the global existence of strong solutions and the unique continuation of weak solutions forNavier-Stokes equations is a still challenging problem.In this paper, combining with the research that we get, I discuss the unique continu-ation for the solution to the given Navier-Stokes equations that has the profound physicalmeaning under some appropriate conditions. In chapter I, we introduce the backgroundknowledge of Navier-Stokes equations as well as previous research results and the mainresults and arrangement of the article. In chapter II, in particular, we introduce variabletransformation of the curl, thus we can reduce the information of the tension item andsimplify the equations. Hence we get equation of the q. In chapter III, in R~n×[0,1], wemainly discuss the equation about q. We use transformation of weighted function in orderto get weight estimation. At the same time, we get the both sides’ estimation of q about tand the estimation of q, through constructing logarithmic convexity of the solution of theequation about q. As you see, we also introduce Appel transformation about q and proveCarleman estimation of equation about(q|~)after the Appel transformation on basis of it. Then,by those lemmas and Carleman estimation above, we mainly stress the unique continuationfor the equation about(q|~)in R~n×[0,1]. This means we accordingly get the uniquecontinuation for the equation about q. At last, with the conditions that have been given, weanalyze the unique continuation for the given Navier-Stokes equations about u in R~n×[0,1].