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Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system
Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion
1. | School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, Guangdong, China |
2. | Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education, and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China |
3. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China |
In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a $(L^2,L^2)$ pullback $\mathscr{D}_μ$ -attractor of this model. Then we show that the pullback $\mathscr{D}_μ$ -attractor attract the $\mathscr{D}_μ$ class (especially all $L^2$ -bounded set) in $L^{2+δ}$-norm for any $δ∈[0,∞)$. Moreover, the solution of the model is shown to be continuous in $H^s$ with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the $(L^2,L^2)$ pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of $\mathscr{D}_{μ}$ in $H^s$ -norm, and thus the existence of a $(L^2, H^s)$ pullback $\mathscr{D}_μ$ -attractor is obtained.
References:
[1] |
R. A. Adams,
Sobolev Spaces, New York: Academic Press, 1975. |
[2] |
M. Anguiano, P. Marín-Rubio and J. Real,
Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.
doi: 10.1016/j.jmaa.2011.05.046. |
[3] |
D. Applebaum,
Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
[4] |
H. Bahouri, J. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[6] |
C. Bardos, P. Penel, U. Frisch and P.-L. Sulem,
Modified dissipativity for a nonlinear evolution equations arising in turbulence, Arch. Rational Mech. Anal., 71 (1979), 237-256.
doi: 10.1007/BF00280598. |
[7] |
P. Biler, C. Imbert and G. Karch,
Barenblatt profiles for a nonlocal porous medium equation, C. R. Math. Acad. Sci. Paris., 349 (2011), 641-645.
doi: 10.1016/j.crma.2011.06.003. |
[8] |
M. Bonforte, Y. Sire and J. L. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[9] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s,p}$ when $s\to \text{1}$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[10] |
L. A. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
L. A. Caffarelli, F. Soria and J. L. Vázquez,
Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc., 15 (2013), 1701-1746.
doi: 10.4171/JEMS/401. |
[12] |
L. A. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[13] |
D. Cao, C. Sun and M. Yang,
Dynamical for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.
doi: 10.1016/j.jde.2015.02.020. |
[14] |
T. Caraballo, G. Łukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[15] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[16] |
V. V. Chepyzhov and M. I. Vishik,
Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. |
[17] |
J. W. Cholewa and T. Dlotko,
Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000.
doi: 10.1017/CBO9780511526404. |
[18] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Lecture Notes in
Mathematics, vol. 1871, Springer-Verlag, Berlin, 2006, 1–43.
doi: 10.1007/11545989_1. |
[19] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[20] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[21] |
T. Dlotko, M. Kania and C. Sun,
Pseudodifferential parabolic equations; two examples, Topol. Methods Nonlinear Anal., 43 (2014), 463-492.
doi: 10.12775/TMNA.2014.028. |
[22] |
J. Droniou and C. Imbert,
Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331.
doi: 10.1007/s00205-006-0429-2. |
[23] |
N. Jacob,
Pseudo Differential Operators and Markov Processes, Vol. Ⅰ, Ⅱ, Ⅲ, Imperial College Press, London, 2005.
doi: 10.1142/9781860947155. |
[24] |
S. Jarohs and T. Weth,
Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.
doi: 10.3934/dcds.2014.34.2581. |
[25] |
G. Karch, Nonlinear evolution equations with anomalous diffusion, Qualitative Properties
of Solutions to Partial Differential Equations, J. Nečas Center for Mathematical Modeling,
Charles University, Prague, 5 (2009), 25–68. |
[26] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[27] |
P. E. Kloeden and M. Rasmussen,
Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011.
doi: 10.1090/surv/176. |
[28] |
G. Lukaszewicz,
On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[29] |
G. Lukaszewicz,
On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.
doi: 10.1016/j.na.2010.03.023. |
[30] |
V. I. Mazya and T. O. Shaposhnikova,
On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[31] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[32] |
J. C. Robinson,
Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0. |
[33] |
C. Sun and Y. Yuan,
$L^p$
-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.
doi: 10.1017/S0308210515000177. |
[34] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[35] |
H. Triebel,
Theory of Function Spaces, Monographs in Mathematics, 78. Birkhuser Verlag, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[36] |
T. Trujillo and B. Wang,
Continuity of strong solutions of reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[37] |
E. Valdinoci,
From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.
|
[38] |
J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear partial
differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen,
Springer, 2012,271–298.
doi: 10.1007/978-3-642-25361-4_15. |
[39] |
J. L. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian
operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[40] |
B. Wang,
Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
show all references
References:
[1] |
R. A. Adams,
Sobolev Spaces, New York: Academic Press, 1975. |
[2] |
M. Anguiano, P. Marín-Rubio and J. Real,
Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.
doi: 10.1016/j.jmaa.2011.05.046. |
[3] |
D. Applebaum,
Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
[4] |
H. Bahouri, J. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[6] |
C. Bardos, P. Penel, U. Frisch and P.-L. Sulem,
Modified dissipativity for a nonlinear evolution equations arising in turbulence, Arch. Rational Mech. Anal., 71 (1979), 237-256.
doi: 10.1007/BF00280598. |
[7] |
P. Biler, C. Imbert and G. Karch,
Barenblatt profiles for a nonlocal porous medium equation, C. R. Math. Acad. Sci. Paris., 349 (2011), 641-645.
doi: 10.1016/j.crma.2011.06.003. |
[8] |
M. Bonforte, Y. Sire and J. L. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[9] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s,p}$ when $s\to \text{1}$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[10] |
L. A. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
L. A. Caffarelli, F. Soria and J. L. Vázquez,
Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc., 15 (2013), 1701-1746.
doi: 10.4171/JEMS/401. |
[12] |
L. A. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[13] |
D. Cao, C. Sun and M. Yang,
Dynamical for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.
doi: 10.1016/j.jde.2015.02.020. |
[14] |
T. Caraballo, G. Łukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[15] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[16] |
V. V. Chepyzhov and M. I. Vishik,
Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. |
[17] |
J. W. Cholewa and T. Dlotko,
Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000.
doi: 10.1017/CBO9780511526404. |
[18] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Lecture Notes in
Mathematics, vol. 1871, Springer-Verlag, Berlin, 2006, 1–43.
doi: 10.1007/11545989_1. |
[19] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[20] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[21] |
T. Dlotko, M. Kania and C. Sun,
Pseudodifferential parabolic equations; two examples, Topol. Methods Nonlinear Anal., 43 (2014), 463-492.
doi: 10.12775/TMNA.2014.028. |
[22] |
J. Droniou and C. Imbert,
Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331.
doi: 10.1007/s00205-006-0429-2. |
[23] |
N. Jacob,
Pseudo Differential Operators and Markov Processes, Vol. Ⅰ, Ⅱ, Ⅲ, Imperial College Press, London, 2005.
doi: 10.1142/9781860947155. |
[24] |
S. Jarohs and T. Weth,
Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.
doi: 10.3934/dcds.2014.34.2581. |
[25] |
G. Karch, Nonlinear evolution equations with anomalous diffusion, Qualitative Properties
of Solutions to Partial Differential Equations, J. Nečas Center for Mathematical Modeling,
Charles University, Prague, 5 (2009), 25–68. |
[26] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[27] |
P. E. Kloeden and M. Rasmussen,
Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011.
doi: 10.1090/surv/176. |
[28] |
G. Lukaszewicz,
On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[29] |
G. Lukaszewicz,
On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.
doi: 10.1016/j.na.2010.03.023. |
[30] |
V. I. Mazya and T. O. Shaposhnikova,
On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[31] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[32] |
J. C. Robinson,
Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0. |
[33] |
C. Sun and Y. Yuan,
$L^p$
-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.
doi: 10.1017/S0308210515000177. |
[34] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[35] |
H. Triebel,
Theory of Function Spaces, Monographs in Mathematics, 78. Birkhuser Verlag, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[36] |
T. Trujillo and B. Wang,
Continuity of strong solutions of reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[37] |
E. Valdinoci,
From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.
|
[38] |
J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear partial
differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen,
Springer, 2012,271–298.
doi: 10.1007/978-3-642-25361-4_15. |
[39] |
J. L. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian
operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[40] |
B. Wang,
Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
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