In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter 2 † ", 0 , where † ", 0 D f 2 C j j arg j Ä ", j j 0 g.0 < " < =2, 0 > 0/, and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < " < =2 and 0 > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large 0 > 0 for given 0 < " < =2. We also prove the maximal L p L q regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x
In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R. In order to prove the decay properties, we first show that the zero points λ± of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: λ± = ±icwhere cg > 0 is the gravitational acceleration and ξ ′ ∈ R N−1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing λ± and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N − 1)-dimensional heat kernel and F −1is the inverse Fourier transform with respect to ξ ′ . However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.
The aim of this paper is to show the existence of -bounded solution operator families for a generalized resolvent problem on the half-space arising from a compressible fluid model of Korteweg type. Our main result covers not only the large resolvent parameter ∈ C + ∶= {z ∈ C|ℜz > 0} so that | |≥ for some > 0, but also small satisfying | |≤ .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.