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Regularity criterion for 3D Navier-Stokes equations in Besov spaces
Well-posedness of abstract distributed-order fractional diffusion equations
1. | School of Mathematics and Statistics, Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi'an Jiaotong University, Xi'an 710049, China, China, China |
References:
[1] |
D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511755323. |
[2] |
R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks, Phys. Rev. E, 51 (1995), 4807-4822.
doi: 10.1103/PhysRevE.51.4807. |
[3] |
E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001. |
[4] |
Alfred. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878.
doi: 10.1090/S0002-9947-1991-1018572-4. |
[5] |
Aleksei. V. Chechkin, R. Gorenflo and Igor. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129.
doi: 10.1103/PhysRevE.66.046129. |
[6] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[7] |
G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[8] |
N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers, J. Func. Anal., 256 (2009), 1387-1407.
doi: 10.1016/j.jfa.2008.07.020. |
[9] |
A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956. |
[10] |
R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. |
[11] |
T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966.
doi: 10.1007/978-3-642-53393-8. |
[12] |
A. Klemm, Hans. P. Müller and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E, 55 (1997), 4413-4422.
doi: 10.1103/PhysRevE.55.4413. |
[13] |
M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, Birkh$\ddota$uzer, 2006.
doi: 10.1007/3-7643-7698-8. |
[14] |
M. Li, C. Chen and F. Li, On fractional powers of generators of fractional resolvent families, J. Func. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[15] |
K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 63 (2011), 1398-1404.
doi: 10.1016/j.camwa.2011.02.038. |
[16] |
F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput., 187 (2007), 295-305.
doi: 10.1016/j.amc.2006.08.126. |
[17] |
F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192. |
[18] |
Rosario. N. Mantegna and Harry. E. Stanley, Stochastic process with ultraslow convergence to a gaussian: the truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.
doi: 10.1103/PhysRevLett.73.2946. |
[19] |
Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001. |
[20] |
Mark. M. Meerschaert and Hans. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235.
doi: 10.1016/j.spa.2006.01.006. |
[21] |
Mark. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216-228.
doi: 10.1016/j.jmaa.2010.12.056. |
[22] |
R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[23] |
Elliott. W. Montroll and George. H. Weiss, Random walks on lattices. II, J. Math. Phys., 167 (1965), 167-181.
doi: 10.1063/1.1704269. |
[24] |
M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32.
doi: 10.1142/S0218348X04002410. |
[25] |
I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999. |
[26] |
J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[27] |
Lewis. F. Richardson, Atmospheric diffusion shown on a distance-neighbout graph, Proc. Roy. Soc., 110 (1926), 709-737.
doi: 10.1098/rspa.1926.0043. |
[28] |
Yakov. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random medium, Theor. Prob. Appl., 27 (1982), 256-268.
doi: 10.1137/1127028. |
[29] |
Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. |
[30] |
G. Zumofen and J. Klafter, Spectral random walk of a single molecule, Chem. Phys. Lett., 219 (1994), 303-309.
doi: 10.1016/0009-2614(94)87062-4. |
show all references
References:
[1] |
D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511755323. |
[2] |
R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks, Phys. Rev. E, 51 (1995), 4807-4822.
doi: 10.1103/PhysRevE.51.4807. |
[3] |
E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001. |
[4] |
Alfred. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878.
doi: 10.1090/S0002-9947-1991-1018572-4. |
[5] |
Aleksei. V. Chechkin, R. Gorenflo and Igor. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129.
doi: 10.1103/PhysRevE.66.046129. |
[6] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[7] |
G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[8] |
N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers, J. Func. Anal., 256 (2009), 1387-1407.
doi: 10.1016/j.jfa.2008.07.020. |
[9] |
A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956. |
[10] |
R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. |
[11] |
T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966.
doi: 10.1007/978-3-642-53393-8. |
[12] |
A. Klemm, Hans. P. Müller and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E, 55 (1997), 4413-4422.
doi: 10.1103/PhysRevE.55.4413. |
[13] |
M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, Birkh$\ddota$uzer, 2006.
doi: 10.1007/3-7643-7698-8. |
[14] |
M. Li, C. Chen and F. Li, On fractional powers of generators of fractional resolvent families, J. Func. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[15] |
K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 63 (2011), 1398-1404.
doi: 10.1016/j.camwa.2011.02.038. |
[16] |
F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput., 187 (2007), 295-305.
doi: 10.1016/j.amc.2006.08.126. |
[17] |
F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192. |
[18] |
Rosario. N. Mantegna and Harry. E. Stanley, Stochastic process with ultraslow convergence to a gaussian: the truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.
doi: 10.1103/PhysRevLett.73.2946. |
[19] |
Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001. |
[20] |
Mark. M. Meerschaert and Hans. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235.
doi: 10.1016/j.spa.2006.01.006. |
[21] |
Mark. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216-228.
doi: 10.1016/j.jmaa.2010.12.056. |
[22] |
R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[23] |
Elliott. W. Montroll and George. H. Weiss, Random walks on lattices. II, J. Math. Phys., 167 (1965), 167-181.
doi: 10.1063/1.1704269. |
[24] |
M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32.
doi: 10.1142/S0218348X04002410. |
[25] |
I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999. |
[26] |
J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[27] |
Lewis. F. Richardson, Atmospheric diffusion shown on a distance-neighbout graph, Proc. Roy. Soc., 110 (1926), 709-737.
doi: 10.1098/rspa.1926.0043. |
[28] |
Yakov. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random medium, Theor. Prob. Appl., 27 (1982), 256-268.
doi: 10.1137/1127028. |
[29] |
Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. |
[30] |
G. Zumofen and J. Klafter, Spectral random walk of a single molecule, Chem. Phys. Lett., 219 (1994), 303-309.
doi: 10.1016/0009-2614(94)87062-4. |
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