具有部分粘性和阻尼的三维热带气候模型的全局正则性.pdf

应用数学MATHEMATICA APPLICATA2024,37(1):80-88Global Regularity for the 3D TropicalClimate Model with Partial Viscosityand DampingZENG Ying(曾颖),BIE Qunyi(别群益)(College of Science,China Three Gorges University,Yichang 443002,China)Abstract:In this paper,we study the three dimensional tropical climate model withdamping|u|1u on the barotropic mode of the velocity.By using the energy estimation,we obtain the global existence and uniqueness of a strong solution when 4 and 32.Key words:Tropical climate model;Global well-posedness;Strong solution;DampingCLC Number:O175.2AMS(2010)Subject Classification:35Q35;76D03Document code:AArticle ID:1001-9847(2024)01-0080-091.IntroductionIn this paper,we consider the following three dimensional tropical climate model withpartial viscosity and damping:tu+(u )u hu+|u|1u+(w w)+p=0,x R3,t 0,tw+(u )w+()w+(w )u+=0,t+(u )+w=0,u=0,u(x,0)=u0(x),w(x,0)=w0(x),(x,0)=0(x),(1.1)where vector fields u(x,t)and w(x,t)denote the barotropic mode and the first baroclinicmode of the velocity field,respectively,p=p(x,t)and=(x,t)stand for the scalar pressureand scalar temperature.The real parameters,and are non-negative constants.Thedamping coefficient 0 and,1 are constants.Here,h:=21+22is the Laplaceoperator in the horizontal variables.System(1.1)is related to the following classical tropicalclimate model with damping:tu+(u )u u+|u|1u+(w w)+p=0,x R3,t 0,tw+(u )w w+(w )u+=0,t+(u )+w=0,u=0,u(x,0)=u0(x),w(x,0)=w0(x),(x,0)=0(x).(1.2)When there is no damping in the system(1.2),i.e.,=0,the global regularity re-sults have been investigated in 1-3 etc.When,=0,the system(1.2)reduces toReceived date:2022-11-28Foundation item:Supported by the National Natural Science Foundation of China(11871305)Corresponding author:BIE Qunyi,male,Han,Hubei,professor,major in partial differentialequation.No.1ZENG Ying,et al.：Global Regularity for the 3D Tropical Climate Model81the original tropical climate model derived by Frierson,Majda and Pauluis4.In this case,YE and ZHU56considered the global strong solutions when the viscosity depends on thetemperature.When 0 in(1.2),MA and WAN7showed the global well-posedness for the smalldata.YUAN and CHEN8obtained the global existence and uniqueness of the strong solutionprovided that 4 and(u0,w0,0)H1(R3)(H1(R3)L2m(R3)(H1(R3)L2m(R3)(1.3)with m 32.On the other hand,when there are damping terms 2|w|1w and 3|1 in(1.2)2and(1.2)3,respectively,YUAN and ZHANG9proved that there exists the global and uniquesolution when,and satisfy some conditions.Berti,Bisconti and Catania10establisheda regularity criterion,thanks to which the local smooth solution(u,w,)can actually beextended globally in time.For some global regularity results for the NavierStokes equations,micropolar equationsand MHD equations with damping terms,we can refer to 11-14.Let us mention that in 14,LIU et al.studied the existence and uniqueness of a global solution for the three-dimensionalBoussinesq-MHD equations with partial viscosity and damping.Motivated by the works mentioned above,we will consider the system(1.1),which is thecase for the system(1.2)with partial viscosity and damping.We focus on the global existenceand uniqueness of the strong solution to the system(1.1).The work of this paper has two main points.First,compared with the system(1.2)considered by 7-8,we investigate the case that(1.1)1contains the Laplace operator huonly with respect to the horizontal variables.Second,we do not need the initial conditionsw0 L2m(R3)and 0 L2m(R3)in(1.3).However,in order to control the term 3u443L2ininequalities(2.12)and(2.14)below,we need the condition 32.Before stating our main theorem,we shall introduce some notations and functional set-tings used in the sequel.Since the horizontal variable xh:=(x1,x2)plays a different role withthe vertical variable x3,it is natural to introduce the so called anisotropic Sobolev spacesHs,sfor all s,s R,that is,the space Hs,sis the Sobolev space with regularity Hsin xhand Hsin x3.It is easy to check that L2=H0,0and H1=H1,1.Moreover,we shall use thenotation h:=(1,2).Definition 1.1We call the function(u(x,t),w(x,t),(x,t)the strong solution of thesystem(1.1)if it is a weak solution of(1.1)satisfyingu(x,t)Lloc(R+;H0,1(R3)L2loc(R+;H1(R3)L+1loc(R+;L+1loc(R3),w(x,t)Lloc(R+;H1(R3)L2loc(R+;H+1(R3),(x,t)Lloc(R+;H1(R3)L2loc(R+;H2(R3),(u,w,)2L2+2t0(hu2L2+w2L2+2L2+u+1L+1)dx(u0,w0,0)2L2,u=0,u(x,0)=u0(x),w(x,0)=w0(x),(x,0)=0(x).(1.4)Now,we state our main result as follows.82MATHEMATICA APPLICATA2024Theorem 1.1 Let u0 H0,1(R3),w0 H1(R3),0 H1(R3)such that divu0=0,4and 32.Then,the system(1.1)admits a unique global solution(u(x,t),w(x,t),(x,t)satisfyingu(x,t)Lloc(R+;H0,1(R3)L2loc(R+;H1(R3)L+1loc(R+;L+1loc(R3),w(x,t)Lloc(R+;H1(R3)L2loc(R+;H+1(R3),(x,t)Lloc(R+;H1(R3)L2loc(R+;H2(R3).Moreover,the solution(u(x,t),w(x,t),(x,t)depends continuously on the initial data.2.Proof of Theorem 1.1This section is devoted to the proof of Theorem 1.1.By a priori estimates and takinglimits of the Galerkin approximated solutions,we can obtain the existence of a global strongsolution.Therefore,in the sequel,we give a priori estimates and prove Theorem 1.1 bystandard procedure.We split this section into two parts.The first part is for the existence while the secondone is for the uniqueness.Step 1A priori estimatesFirst,multipling(1.1)1,(1.1)2and(1.1)3by u,w and,respectively,and summing themup,we get after integrating by parts that12ddt(u2L2+w2L2+2L2)+hu2L2+w2L2+2L2+u+1L+1=R3(u )u udx R3(u )w wdx R3(u )dx R3p udxR3 (w w)udx R3(w )u w R3 wdx R3 w dx=0.(2.1)Integrating with respect to t,we arrive at(u,w,)2L2+2t0(hu(t)2L2+w(t)2L2+(t)2L2+u(t)+1L+1)d=(u0,w0,0)2L2.(2.2)Thus,it holds thatu(x,t)L(0,T;L2(R3)L2(0,T;H1,0(R3)L+1(0,T;L+1(R3),w(x,t)L(0,T;L2(R3)L2(0,T;H(R3),(x,t)L(0,T;L2(R3)L2(0,T;H1(R3).(2.3)Next,we shall obtain estimates of the derivative of(u,w,).To this end,we multiply(1.1)1by 23u and use integration by parts over R3to get12ddt3u(t)2L2+h3u(t)2L2R3(u )u 23u(t)dx R3|u|1u 23u(t)dx R3 (w w)23u(t)dx=0.(2.4)Similarly,we deduce from(1.1)2and(1.1)3that12ddtw2L2+1w2L2R3(u)wwdxR3(w)uwdxR3wdx=0(2.5)No.1ZENG Ying,et al.：Global Regularity for the 3D Tropical Climate Model83and12ddt(t)2L2+(t)2L2R3(u )dx R3(w)dx=0.(2.6)Summing up(2.4),(2.5)and(2.6),we obtain12ddt(3u2L2+w2L2+2L2)+h3u2L2+1w2L2+2L2=R3(u )u 23u(t)dx+R3|u|1u 23u(t)dx+R3 (w w)23u(t)dx+R3(u )w w(t)dx+R3(w )u wdx+R3 wdx+R3(u )dx+R3(w)dx:=8i=1Ii.(2.7)It is the position to estimate the terms on the right hand of(2.7).First,using the H olderinequality and Gagliardo-Nirenberg inequality,we have|I1|=?3k,l=1R33ukkul3uldx?=?2k=13l=1R33ukkul3uldx 3l=1R3(1u1+2u2)3ul3uldx?=?2k=13l=1R33ukulk3uldx?+2?2k=13l=1R3uk3ul3kuldx?:=I11+I12,(2.8)where we use the fact 3u3=1u1 2u2.Next,we estimate I11(t)and I12(t).For 3,one has|I11|2k=13l=1R3|ul|3uk|21|3uk|31|k3ul|dx2k=13l=1ul|3uk|21L1|3uk|31L2(1)3k3ulL24|u|12|3u|2L2+12h3u2L2+C3u2L2.(2.9)Regarding I12(t),applying the H older and Young inequalities,we get for 3 that|I12|4|u|12|3u|2L2+12h3u2L2+C3u2L2.(2.10)For I2(t),applying integration by parts,we obtainI2=R3|u|1u 23u(t)dx=R3|u|1|3u(t)|2dx 2R33|u(t)|23|u(t)|1dx|u|123u2L24(1)(+1)23|u|+122L2.(2.11)84MATHEMATICA APPLICATA2024With respect to I3(t),when 32,using integration by parts,the H older and Young inequal-ities,one deduces|I3|=?R3 (w w)23u(t)dx?=R3w w 23u(t)dx+R3w w 23u(t)dx CR3(|3w|w|3u(t)|+|w|3w|3u(t)|)dx C3uL2w2L4+C3uL2wL3wL6 C3uL2+1w32L2w432L2+1w2L2+Cw2L23u443L2+1w2L2+Cw2L2w22L23u443L2+1w2L2+C(w2L2+w2L2)3u443L2+1w2L2+C(w2L2+w2L2)(1+3u2L2)+1w2L2+C(w2L2+w2L2)3u2L2+C(w2L2+w2L2).(2.12)For I4(t),applying the Sobolev embedding,H older and Gagliardo-Nirenberg inequalities,wehave|I4|=?R3(u )w w(t)dx?uL2wL3wL6 CuL2w4522L2+1w122L2 CuL2+1w52L2w452L2+1w2L2+Cu445L2w2L2.(2.13)Similiar to the estimation of I3(t),it follows that|I5|=?R3(w )u wdx?R3|w|hu|w|dx+R3|w|3u|w|dx huL2wL6wL3+3uL2wL6wL3 ChuL2w432L2+1w32L2+C3uL2w432L2+1w32L2+1w2L2+Chu443L2w2L2+Cw2L2w22L23u443L2+1w2L2+C(1+hu2L2)w2L2+C(w2L2+w2L2)(1+3u2L2).(2.14)For I6(t),using the Young and Gagliardo-Nirenberg inequalities,we deduce|I6|=?R3 wdx?L2wL2 L2+1w2+1L2w1+1L2+1w2L2+C+1L2w1L2+1w2L2+C(1+2L2)w1L2+1w2L2+C2L2w1L2+Cw1L2.(2.15)With respect to I7(t),applying the H older,Young and Gagliardo-Nirenberg inequalities,wehave for 4 that|I7|=?R3(u )dx?No.1ZENG Ying,et al.：Global Regularity for the 3D Tropical Climate Model85 uL+1L2L2(+1)1 uL+1+4+1L22+1L2 2L2+Cu2(+1)2L+12L2 2L2+C(1+u+1L+1)2L2.(2.16)For I8(t),using the Young inequality,we get|I8|=?R3(w)dx?2L2+Cw2L2.(2.17)Substituting Ii(i=1,2,8)into(2.7),we haveddt(3u(t)2L2+w(t)2L2+(t)2L2)+h3u(t)2L2+1w(t)2L2+(t)2L2+|u|123u2L2+3|u|+122L2C(1+u445L2+u+1L+1+hu2L2+w2L2+w2L2+w1L2)(3u(t)2L2+w(t)2L2+(t)2L2).(2.18)Applying Gronwalls inequality to(2.18),we obtain that3u(t)2L2+w(t)2L2+(t)2L2+t0(h3u(t)2L2+1w(t)2L2+(t)2L2)d C(t,u0,w0,0)(2.19)andu(x,t)L(0,T;H0,1(R3)L2(0,T;H1(R3),w(x,t)L(0,T;H1(R3)L2(0,T;H+1(R3),(x,t)L(0,T;H1(R3)L2(0,T;H2(R3).(2.20)Strictly speaking,the a priori estimates in Step 1 is applicable to the approximate solutionconstructed by the Friederichs regularization method(see 15,Chapter 6).Therefore,it canonly be transmitted to the limit in the sequence of solutions.We focus on the limit ofthe nonlinear terms.Through Step 1,we have un L2(0,T;L2(R3).Using the Rellich-Kondrachov theorem in 16,we obtain that unis the strong convergence in L2(0,T;L2(R3).Let be a test function.We haveT0(un)un(u )u,ds=T0(un u)un,+T0(u )(un u),ds=T0(un u)un,T0(u ),(un u)dsCT0un uL2unL2ds+CT0uL2un uL2ds 0,as n .So thatT0(un)un,ds T0(u )u,ds.For damping term|u|1u,inspired byTheorem 1 in 11 andT0R3|u|+1dxds C,we get that un u strongly in Lp(0,T;Lp(R3)for 2 p 1.In summary,we can getT0|un|1un,ds T0|u|1u,ds.86MATHEMATICA APPLICATA2024For all the other nonlinear terms,we get the convergence by using the similar method.Thus,the existence of solution to the system(1.1)is proved.Step 2UniquenessIt remains to prove the uniqueness of the solution.Let(u,w,)and(u,w,)be twosolutions associated to the initial data(u0,w0,0)and(u0,w0,0),respectively,satisfying theconditions in Theorem 1.1.Let(f,g,q)=(u u,w w,),then(f,g,q)satisfiestf+u f+f u hf+|u|1u|u|1 u+(w g)(w g)+p=0,tg+u g+f w+()g+w f+g u+q=0,tq+u q+f q+g=0.(2.21)Now,multiplying(2.21)1,(2.21)2and(2.21)3by f,g and q,respectively,integrating overR3,and summing them up,we get after integrating by parts that12ddt(f2L2+g2L2+q2L2)+hf2L2+g2L2+q2L2+R3(|u|1u|u|1 u)(u u)dx=R3u f fdx R3f u fdx R3 (w g)fdx R3 (w g)fdxR3u g gdx R3f w gdx R3w f gdx R3g u gdxR3q gdx R3u q qdx R3f qdx R3 g qdx=R3f u fdx R3 (w g)fdx R3 (w g)fdx R3f w gdxR3w f gdx R3g u gdx R3f qdx:=7k=1Jk.(2.22)First,the last term on the left hand of(2.22)is handled asR3(|u|1u|u|1 u)(u u)dx (uL+1 uL+1)(uL+1 uL+1)0.(2.23)Making use of the H older,Young and Gagliardo-Nirenberg inequalities,one has|J1|=R3f u fdx=2k=13l=1R3fkk ulfldx+3l=1R3f33 ulfldx2k=13l=1RfkL4hk ulL2hflL4hdx3+3l=1Rf3L2h3 ulL4hflL4hdx3ChuLfL2hfL2fL4h+Cf3LfL2h3uL2fL4hfL2fL4hhf2L2+C(3hu2L2+hu2L2+3u2L2)f2L2,(2.24)where we have used the following inequalities:2L2fL4h CRL2hhL2hdx3 CL2hL2and3f3L2 ChfL2,No.1ZENG Ying,et al.：Global Regularity for the 3D Tropical Climate Model87due to 3f3=2f2 1f1.For J2(t)and J3(t),using the H older,Young and Gagliardo-Nirenberg inequalities,weget|J2|+|J3|2k=13l=1R3(w g)kfldx+3l=1R3(w g)3fldx+2k=13l=1R3(w g)kfldx+3l=1R3(w g)3fldxgL4wL4(hfL2+3fL2)+gL4 wL4(hfL2+3fL2)gL4(wL4+wL4)(hfL2+3fL2)C(hf2L2+3f2L2)+Cg432L2g32L2(w432L2w32L2+w432L2 w32L2)hf2L2+C3(u u)2L2+Cg432L2g32L2(w432L2w32L2+w432L2 w32L2)hf2L2+C3u2L2+C3 u2L2+g2L2+Cg2L2(w643L2+w643L2).(2.25)For J4(t),applying the H older,Young and Gagliardo-Nirenberg inequalities,we infer|J4|CfL2gL4 wL4CfL2g34L2g434L2+1 w34L2 w434L2Cg32L2g432L2+Cf2L2+1 w32L2 w432L2g2L2+Cg2L2+C(+1 w2L2+w2L2)f2L2.(2.26)Arguing similarly as the estimates of J2(t)and J3(t),one has|J5|hf2L2+C3u2L2+C3 u2L2+g2L2+Cg2L2w643L2.(2.27)Using the H older and Gagliardo-Nirenberg inequalities,we have|J6|=?R3g u gdx?R3|g|h u|g|dx+R3|g|3 u|g|dxg2L4h uL2+g2L43 uL2g432L2g32L2h uL2+g432L2g32L23 uL2g2L2+Cg2L2(h u443L2+3 u443L2)g2L2+Cg2L2(2+h u2L2+3 u2L2)(2.28)and|J7|R3|f|q|dx fL2qL6L3CfL2qL212L212L2q2L2+C(2L2+2L2)f2L2.(2.29)SetF(t)=(1+3h u2L2+h u2L2+3 u2L2+3u2L2+w2L2+w643L2+w643L2+1 w2L2+2L2+2L2).88MATHEMATICA APPLICATA2024Inserting estimates(2.23)-(2.29)into(2.22)yields thatddt(f,g,q)2L2 CF(t)(f,g,q)2L2.(2.30)By virtue of(2.19)and(2.20),it holds that F(t)L1loc(R+).Therefore,applying Gronwallsinequality to(2.30)leads to the uniqueness.The proof of Theorem 1.1 is completed.References:1 LI J,TITI E S.A tropical atmosphere model with moisture:global well-posedness and relaxationlimitJ.Nonlinearity,2016,29(9):2674.2 YE Z.Global regularity for a class of 2D tropical climate modelJ.J.Math.Anal.Appl.,2017,446(1):307-321.3 DONG B Q,WU J,YE Z.Global regularity for a 2D 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criterion for a 3D tropical climate model withdampingJ.J.Math.Anal.Appl.,2023,518(1):126685.11 CAI X,JIU Q.Weak and strong solutions for the incompressible NavierStokes equations withdampingJ.J.Math.Anal.Appl.,2008,343(2):799-809.12 YE Z.Global existence of strong solution to the 3D micropolar equations with a damping termJ.Appl.Math.Letters,2018,83:188-193.13 ZHANG Z,WU C,YAO Z.Remarks on global regularity for the 3D MHD system with dampingJ.Appl.Math.Comput.,2018,333:1-7.14 LIU H,DENG H,LIN L,et al.Well-posedness of the 3D Boussinesq-MHD equations with partialviscosity and dampingJ.J.Math.Anal.Appl.,2022,515(2):126437.15 BAHOURI H,CHEMIN J Y,DANCHIN R.Fourier Analysis and Nonlinear Partial DifferentialEquationsM.Berlin:Springer,2011.16 ADAMS R A.The Rellich-Kondrachov theorem for unbounded domainsJ.Arch.Ration.Mech.Anal.,1968,29(5):390-394.具有部分粘性和阻尼的三维热带气候模型的全局正则性曾颖,别群益(三峡大学理学院,湖北 宜昌 443002)摘要：本文研究在速度的正压模态上具有阻尼|u|1u的三维热带气候模型.利用能量估计方法,我们得到了当 4和 32时该模型强解的全局存在性和唯一性.关键词：热带气候模型;全局适定性;强解;阻尼