(untitled) (Journal of Partial Differential Equations, 2024)
We study the regularity and convergence of solutions for the n-dimensional (n=2,3) fourth-order vector-valued Helmholtz equationsequation*VFHEVFHE u-β Δ u+γ (-Δ)^2 u =v equation* for a given v in several Sobolev spaces, where β>0 and γ>0 are two given constants. By making use of the Fourier multiplier theorem, we establish the regularity and the L^p-L^q estimates of solutions for Eq. (VFHE) under the condition vL^p(R^n). We then derive the convergence that a solution u of Eq. (VFHE) approaches v weakly in L^p(R^n) and strongly in L^q(R^n) as the parameter pair ( β,γ) approaches (0,0). In particular, as an application of the above results, for (v,u) solving the following viscous incompressible fluid equationsalign*INSINS cases v_t+u· abla v+v· abla u^T+ abla p= u Δ v, div v=div u=0, cases align*we gain the strong convergence in L^∞([0,T],L^s(R^n)) from the Eqs. (VFHE)-(INS) to the Navier-Stokes equations as the parameter pair (β,γ) tending to (0,0), where s=2hh-2 with h>n.