一类非局部Cahn-Hilliard方程弱解的存在唯一性
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摘要
研究一类对流非局部Cahn-Hilliard方程的Neumann问题.通过一致Schauder估计和Leray-Schauder不动点定理,得到了该问题经典解的存在唯一性.进而,利用弱收敛方法得到了该问题弱解的存在唯一性.
This paper studies a nonlocal convective Cahn-Hilliard equation with Neumann boundary condition. Based on the uniform Schauder estimates and Leray-Schaefer ?xed point theorem, we obtain the existence and uniqueness of classical solutions. And then, by continuous method, we get the existence and uniqueness of weak solutions.
This paper studies a nonlocal convective Cahn-Hilliard equation with Neumann boundary condition. Based on the uniform Schauder estimates and Leray-Schaefer ?xed point theorem, we obtain the existence and uniqueness of classical solutions. And then, by continuous method, we get the existence and uniqueness of weak solutions.
引文
[1]Colli P,Frigeri S,Grasselli M.Global existence of weak solutions to a nonlocal Cahn-HilliardNavier-Stokes system[J].J.Math.Anal.Appl.,2012,386:428-444.
[2]Frigeri S,Grasselli M.Krejc′?P,Strong solutions for two-dimensional nonlocal Cahn-Hilliard-NavierStokes systems[J].J.Differential Equations,2013,255:2587-2614.
[3]Frigeri S,Grasselli M.Global and trajectories attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system[J].J.Dynam.Differential Equations,2012,24:827-856.
[4]Frigeri S,Grasselli M.Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials[J].Dyn.Partial Differ.Equ.,2012,9:273-304.
[5]Kwek K H.On the Cahn-Hilliard type equation[D].Atlanta:Georgia Institute of Technology,1991.
[6]Liu C.On the convective Cahn-Hilliard equation with degenerate mobility[J].J.Math.Anal.Appl.,2008,344:123-144.
[7]Zaks M A,Podolny A,Nepomnyashchy A A,et al.Periodic stationary patterns governed by a convective Cahn-Hilliard equation[J].Siam J.Appl.Math.,2005,66:700-720.
[8]Cahn J W,Hilliard J E.Free energy of a nonuniform system I,interfacial free energy[J].J.Chen.Phys.,1958,28:258-267.
[9]Golovin A A,Davis S H,Nepomnyashchy A A.A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth[J].Phys.D.,1998,122:202-230.
[10]Bates P W,Han J.The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation[J].J.Math.Anal.Appl.,2005,311:289-312.
[11]Bates P W,Han J.The Neumann boundary problem for a nonlocal Cahn-Hilliard equation[J].J.Differential Equations,2005,212:235-277.
[12]Han J.The cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation[J].Electron.J.Differential Equations,2004,113:1-9.
[13]Giacomin G,Lebowitz J L.Phase segregation dynamics in particle systems with long range interactions.I.Macroscopic limits[J].J.Stat.Phy.,1997,87:37-61.
[14]Giacomin G,Lebowitz J L.Phase segregation dynamics in particle systems with long range interactions.II.Phase motion[J].SIAM J.Appl.Math.,1998,58:1707-1729.
[15]Eden A,Kalantarov V K.The convective Cahn-Hilliard equation[J].Appl.Math.Lett.,2007,20:455-461.
[16]Golovin A A,Nepomnyashchy A A,Davis S H,et al.Convective Cahn-Hilliard models:from coarsening to roughening[J].Phys.Rev.Lett.,2011,86(8):1550-1553.
[17]Gajewski H,Zacharias K.On a nonlocal phase separation model[J].J.Math.Anal.Appl.,2003,286:11-31.
[18]Liu C,Li Z.Existence of solutions for a nonlocal epitaxial thin film growing equation[J].Arch.Math.,2012,99:157-168.
[19]Alikakos N D.Lpbounds of solutions of reaction-diffusion equations[J].Comm.Partial Differential equations,1979,4(8):827-868.
[20]Ladyzhenskaja O A,Solonikov V A,Uralceva N N.Linear and Quasilinear Equations of Parabolic Type[M].Providence:American Mathematical Society,1968.
[2]Frigeri S,Grasselli M.Krejc′?P,Strong solutions for two-dimensional nonlocal Cahn-Hilliard-NavierStokes systems[J].J.Differential Equations,2013,255:2587-2614.
[3]Frigeri S,Grasselli M.Global and trajectories attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system[J].J.Dynam.Differential Equations,2012,24:827-856.
[4]Frigeri S,Grasselli M.Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials[J].Dyn.Partial Differ.Equ.,2012,9:273-304.
[5]Kwek K H.On the Cahn-Hilliard type equation[D].Atlanta:Georgia Institute of Technology,1991.
[6]Liu C.On the convective Cahn-Hilliard equation with degenerate mobility[J].J.Math.Anal.Appl.,2008,344:123-144.
[7]Zaks M A,Podolny A,Nepomnyashchy A A,et al.Periodic stationary patterns governed by a convective Cahn-Hilliard equation[J].Siam J.Appl.Math.,2005,66:700-720.
[8]Cahn J W,Hilliard J E.Free energy of a nonuniform system I,interfacial free energy[J].J.Chen.Phys.,1958,28:258-267.
[9]Golovin A A,Davis S H,Nepomnyashchy A A.A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth[J].Phys.D.,1998,122:202-230.
[10]Bates P W,Han J.The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation[J].J.Math.Anal.Appl.,2005,311:289-312.
[11]Bates P W,Han J.The Neumann boundary problem for a nonlocal Cahn-Hilliard equation[J].J.Differential Equations,2005,212:235-277.
[12]Han J.The cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation[J].Electron.J.Differential Equations,2004,113:1-9.
[13]Giacomin G,Lebowitz J L.Phase segregation dynamics in particle systems with long range interactions.I.Macroscopic limits[J].J.Stat.Phy.,1997,87:37-61.
[14]Giacomin G,Lebowitz J L.Phase segregation dynamics in particle systems with long range interactions.II.Phase motion[J].SIAM J.Appl.Math.,1998,58:1707-1729.
[15]Eden A,Kalantarov V K.The convective Cahn-Hilliard equation[J].Appl.Math.Lett.,2007,20:455-461.
[16]Golovin A A,Nepomnyashchy A A,Davis S H,et al.Convective Cahn-Hilliard models:from coarsening to roughening[J].Phys.Rev.Lett.,2011,86(8):1550-1553.
[17]Gajewski H,Zacharias K.On a nonlocal phase separation model[J].J.Math.Anal.Appl.,2003,286:11-31.
[18]Liu C,Li Z.Existence of solutions for a nonlocal epitaxial thin film growing equation[J].Arch.Math.,2012,99:157-168.
[19]Alikakos N D.Lpbounds of solutions of reaction-diffusion equations[J].Comm.Partial Differential equations,1979,4(8):827-868.
[20]Ladyzhenskaja O A,Solonikov V A,Uralceva N N.Linear and Quasilinear Equations of Parabolic Type[M].Providence:American Mathematical Society,1968.