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

作　　者：LI Jing PENG Weimin WANG Yue 

机构地区：[1]Center for Applied Mathematics,Tianjin University,Tianjin 300072,China [2]College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China

出　　处：《Journal of Partial Differential Equations》2024年第3期309-325,共17页偏微分方程（英文版）

基　　金：supported by the National Natural Science Foundation of China under grants(No.11626156);supported by the Natural Science Foundation of Tianjin(No.20JCYBJC01410).

摘　　要：We study the regularity and convergence of solutions for the n-dimensional(n=2,3)fourth-order vector-valued Helmholtz equations u-βΔu+γ(-Δ)^(2)u=v for a given v in several Sobolev spaces,whereβ>0 andγ>0 are two given constants.By making use of the Fourier multiplier theorem,we establish the regularity and the L_(p)-L_(9)estimates of solutions for Eq.(VFHE)under the condition v∈L_(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 incompressiblefluid equations{div v=div u=0,v(t)+u·△V+V·△u^(T)+·△P=V△V(INS)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=2h/(h-2)withh>n.

关 键 词：Fourier multiplier theorem fourth-order Helmholtz equation REGULARITY CONVERGENCE 

分 类 号：O175.27[理学—数学]