Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application

Abstract

We study the regularity and convergence of solutions for the n-dimensional (n=2,3) fourth-order vector-valued Helmholtz equations

(VFHE)$\mathbf{u}-\beta \Delta \mathbf{u}+\gamma(-\Delta)^{2} \mathbf{u}=\mathbf{v}$

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 $\mathbf{v} \in L^{p}\left(\mathbb{R}^{n}\right)$. We then derive the convergence that a solution u of Eq. (VFHE) approaches v weakly in $L^{p}\left(\mathbb{R}^{n}\right)$ and strongly in $L^{p}\left(\mathbb{R}^{n}\right)$ as the parameter pair (β,γ) approaches (0,0). In particular, as an application of the above results, for $(\mathbf{v}, \mathbf{u})$ solving the following viscous incompressible fluid equations

(INS)$\left\{\begin{array}{l}\mathbf{v}_{t}+\mathbf{u} \cdot \nabla \mathbf{v}+\mathbf{v} \cdot \nabla \mathbf{u}^{T}+\nabla p=v \Delta \mathbf{v} \\\operatorname{div} \mathbf{v}=\operatorname{div} \mathbf{u}=0\end{array}\right.$

we gain the strong convergence in $L^{\infty}\left([T], L^{S}\left(\mathbb{R}^{n}\right)\right)$ from the Eqs. (VFHE)-(INS) to the Navier-Stokes equations as the parameter pair (β,γ) tending to (0,0), where $s=\frac{2 h}{h-2}$ with h>n.

Keywords

Fourier multiplier theorem / fourth-order Helmholtz equation / regularity / conver-gence