Rayleigh-Taylor Instability For Nonhomogeneous Incompressible Viscoelastic Fluids

In this article,we study the R.ayleigh-Taylor instability of a nonhomoge-neous incompressible viscoelastic fluid in the presence of a uniform gravitational field.At present,there exist many mathematical models to describe the motion of incompressible viscoelastic fluids.Here we adopt the Oldroyd-B model to study Rayleigh-Taylor instability.The Oldroyd-B model includes the mass equation,the momentum equation and deformation equation.It's well-known that there exists an unstable solution in the sense of H2-norm for the Rayleigh—Taylor prob-lem of Oldroyd-B model,if the elastic coefficient is less than a critical number.However,we use some mathematical techniques to further give an unstable solu-tion in the sense of L'-norm,if the fluid domain is the horizontal periodic domain with infinite height.Next we briefly introduce our proof ideas:First,we linearize the Oldroyd-B model around the Rayleigh-Taylor equi-librium and get the linearized viscoelastic Rayleigh-Taylor problem.Thus,we can use the method of discrete Fourier transformation to construct an unstable solution to the linearized viscoelastic Rayleigh-Taylor problem.Then,based on the structure of Oldroyd-B model,we develop a new energy function,which includes a integral with respect to t.Thus,for any local solution of viscoelastic Rayleigh-Taylor problem,we can derive from Oldroyd-B model that the new energy function of local solution enjoys a Gronwall-type inequality.This inequality makes sure that the higher order norm of the solution can be controlled by that of L2-norm.Finally,based on the first two steps,we can further use the instability boot?strap argument with more refined method of energy estimates to prove the non-linear instability of the viscoelastic Rayleigh-Taylor problem in the sense of L2-norm.