1、2023 年 6 月第 39 卷 第 2 期纯粹数学与应用数学Pure and Applied MathematicsJun.2023Vol.39 No.2Paralinearization of water waves with a moving bottomSHAO Xinhua1,ZANG Aibin2(1.School of Mathematics,Northwest University,Xian 710127,China;2.School of Mathematics and Computer Science and Center of Applied Mathematics,Yi
2、chun University,Yichun 336000,China)Abstract：In this paper,we study irrotational incompressible water waves with grav-itation and surface tension in a moving domain,where there is a given moving bottombesides a free upper boundary.The main goal of this paper,using paradifferential cacu-lus,is to par
3、alinearize Zakharov formulation in nonlinear water wave problems.Defininga regularized mapping via Possion kernel to flatten the boundary makes the process ofparalinearization more delicate.The paralinearization result makes the nonlinear waterwave equations a linear system,which lays a foundation f
4、or studying the well-posednessof the water waves with a moving bottom.Keywords：water wave problem,Zakharov system,paralinearization,Dirichlet-Neumannoperator,moving bottom condition2010 MSC：76B07,76B15Document Code：AArticle ID：1008-5513(2023)02-0159-27DOI：10.3969/j.issn.1008-5513.2023.02.0011 Introd
5、uction1.1 Formulation of the water waveWe consider an incompressible irrotational gravity fluid with surface tension,whichmoves in a time-dependent domaint=(x,y)Rd R|b(t,x)y (t,x)located between a free surfacet=(x,y)Rd R|y=(t,x)收稿日期：2023-03-01.接收日期：2023-03-16.基金项目：国家自然科学基金(12126359,12261093);江西省自然科学
6、基金(20224ACB201004).作者简介：邵鑫华(1997-),硕士生,研究方向:流体力学方程组的数学理论.作者简介：臧爱彬(1979-),博士,教授,研究方向:非线性偏微分方程.160纯粹数学与应用数学第 39 卷and a moving bottomt=t t=(x,y)Rd R|y=b(t,x)for each t I:=0,T,where d=1 or 2 denotes the spatial dimensions of the fluid,tis the time variable,x and y denote respectively horizontal and vert
7、ical components ofspatial variable,(t,x)is a graph of free surface and b(t,x)is a given time-dependentfunction relating to bottom.For all t I,the fluid lies in the domain=(t,x,y)|t I,(x,y)t,with an upper boundary=(t,x,y)I Rd R|y=(t,x),and a time-dependent bottom=(t,x,y)I Rd R|y=b(t,x).We next make t
8、he following assumptions on domain:at t=0 the domain tsatisfies(Ht)h 0:t(x,y)Rd R:y 0,and g Lloc()equals 1 near the top boundary of.Moreover,for simplicity we assume the density of fluid is constant.The waterwaves problem can be described as the Euler system1tv+v x,yv=x,yP geyint,divx,yv=0int,curlx,
9、yv=0int,t=1+|2v ont,P Patm=H()ont,v nm=dmdt nmont,(1)第 2 期邵鑫华 等:具有移动底边界的水波问题的仿线性化161where =(xi)1id,x,y=(x,y),gey(with the constant g 0)stands forgravity acceleration.H()=div(1+|2)is twice the mean curvature of freesurface t,0 is the surface tension coefficient(without loss of generality,take=1),m=m(
10、t,x)=(x,b(t,x),=11+|2(,1)Tand nm=11+|b|2(b,1)Tdenote respectively the outward-pointing unit normal ontand t,Patmstands for the atmosphere pressure,assumed to be a constant.Thefluid velocity v=(v1,v2,vd,vd+1)with vi=vi(x,y)for i=1,d+1,the scalerpressure P and the graph of surface are unknowns.Equatio
11、n(1)1is the conservationof momentum,and the equations(1)1,2state that the fluid is incompressible and irro-tational respectively.The fourth equation in(1)is the kinematic boundary condition,which means that the free surface moves with the fluid.the equation(1)5is the dy-namic condition corresponding
12、 to a balance of forces across the upper surface.The lastone means that the normal velocity of bottom coincides with that of fluid,and simplecomputation implies that k=dmdt nm=11+|b|2tb.There are many mathematical problems in the fluid mechanics equations,whichhave been studied in many fields,such a
13、s magneto hydrodynamics,water wave prob-lems,boundary layer flow,etc.For example,Gui,Li and Li2established the lineardecay of the strong solutions to the linearized system and obtained H3-regularity ofthe global strong solutions to the system.Sun and Zang3obtained the global well-posedness of the bo
14、undary layer equations in half plane,by deriving Prandtl typeequations from the asymptotic expansions of Euler-equations.Water wave problems,from the classical to the singular,are widely concerned as well.Ming and Wang4proved the local well-posedness for a water-wave problem in a bounded two-dimensi
15、onalcorner domain,where the contact angles are less than/16.On the basic of References5-6,Alazard,Burq and Zuily7proved that the Euler system(1)is equivalent to theZakharov system(2),in the case of fixed bottom.Our paper is to study classical waterproblems,which means that the upper boundary is stri
16、ctly separated from the bottomboundary.We are also able to prove that the Euler system(1)can be equivalentlyreformulated as the Zakharov systemt G(,b)=0,t+g+H()+12|212(+G(,b)21+|2=0,(2)where and are unknowns,G(,b)stands for Dirichlet-Neumann operator(D-N162纯粹数学与应用数学第 39 卷operator for short)defined by(G(,b)(t,x)=1+|2|y=(t,x),=(y)(t,x,(t,x)(t,x)()(t,x,(t,x),(3)where the function is harmonic,which satisfies Possions equation with Dirichlet-Neumann boundary conditions:x,y=0int,=(t,x)ont,n(m)=dmdt nm
