Classic solutions to the equation of local fluctuations of the Riesz gravitational fields and their properties
Abstract
UDC 517.937, 519.21
We consider a pseudodifferential equation involving the Riesz operator of fractional differentiation, which is a natural generalization of the well-known equation of fractal diffusion. Its fundamental solution to the Cauchy problem is the density of probability distribution of local interaction forces for moving objects in the corresponding Riesz gravitational field. For this equation, we establish the correct solvability of the Cauchy problem in the class of unbounded, discontinuous initial functions with an integrable singularity. In addition, the form of the classical solution of this problem is found and its smoothness properties and behavior at infinity are investigated. Moreover, under certain conditions on the fluctuation coefficient, we obtain an analogue of the maximum principle and use it to prove the uniqueness of the solution to the Cauchy problem.
References
M. Riesz, Potentiels de divers ordres et leurs fonctions de Green, C. R. Congr. Intern. Math. Oslo, 2, 62 – 63 (1936).
V. A. Litovchenko, Holtsmark fluctuations of non-stationary gravitational fields, Ukr. Mat. Zh. 73, №1, 69 – 76 (2021); https://doi.org/10.37863/umzh.v73i1.6113. DOI: https://doi.org/10.1007/s11253-021-01909-y
L´evy P., Calcul des probabilities, Gauthier-Villars, Paris (1925).
V. M. Zolotarev, Odnomernye ustojchy`vye raspredeleny`ya, Nauka, Moskva (1983).
J. Holtsmark, Über die Verbreiterung von Spektrallinier, Ann. Phys., 58, 577 – 630 (1919), https://doi.org/10.1002/andp.19193630702 DOI: https://doi.org/10.1002/andp.19193630702
S. Chandrasekhar, Stohastic problems in physics and astronomy, Rev. Modern Phys., 15, № 1, 1 – 89 (1943), https://doi.org/10.1103/RevModPhys.15.1 DOI: https://doi.org/10.1103/RevModPhys.15.1
V. A. Litovchenko, Pseudodifferential Equation of Fluctuations of nonstationary gravitational fields, J. Math., 2021, Article ID 6629780, (2021) 8 p.; https://doi.org/10.1155/2021/6629780. DOI: https://doi.org/10.1155/2021/6629780
S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and derivatives of fractional order and some of their applications, ``Nauka i Tekhnika'', Minsk (1987).
Oliver Ibe, Markov processes for stochastic modeling, 2nd Ed., Elsevier (2013); https://doi.org/10.1016/C2012-0-06106-6. DOI: https://doi.org/10.1016/C2012-0-06106-6
V. V. Uchajky`n, Metod drobnyx proy`zvodnyx, Arty`shok: Ul`yanovsk (2008).
N. Jacob, Pseudo differential operators and Markov processes, in 3 vol. Imperial College Press, London (2001, 2002, 2005), https://doi.org/10.1142/9781860949746 DOI: https://doi.org/10.1142/p245
J. Bertoin, L´evy processes, Cambridge Tracts in Math., vol. 121, Cambridge Univ. Press, Cambridge (1996).
D. Applebaum, L´evy processes and stochastic calculus, Cambridge Univ. Press, Cambridge (2009); https://doi.org/10.1017/CBO9780511809781. DOI: https://doi.org/10.1017/CBO9780511809781
T. A. Agekyan, Teory`ya veroyatnostej dlya astronomov y` fy`zy`kov, Nauka, Moskva (1974).
I. I. Sobel’man, An introduction to the theory of atomic spectra, Int. Ser. Natur. Phil., vol. 40, (1972); https://doi.org/10.1016/C2013-0-02394-8 DOI: https://doi.org/10.1016/C2013-0-02394-8
M. Kacz, Veroyatnost y smezhnye voprosy v fyzyke, Myr, Moskva (1965).
A. F. Nikiforov, V. G. Novikov, V. B. Uvarov, Kvantovo-statisticheskie modeli vysokotemperaturnoj plazmy i metody rascheta rosselandovyx probegov i uravnenyi sostoyaniya, Fizmatlit, Moskva (2000).
C. Bucur, E. Valdinoci, Non-local diffusion and applications, Lec. Notes Unione Mat. Ital., 20 (2016); https://doi.org/10.1007/978-3-319-28739-3. DOI: https://doi.org/10.1007/978-3-319-28739-3
A. Reynolds, Liberating L´evy walk research from the shackles of optimal foraging, Phys. Life Rev., 14, 59 – 83 (2015). DOI: https://doi.org/10.1016/j.plrev.2015.03.002
G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, Lévy flights in random searches, Phys. A, 282, № 1-2, 1 – 12 (2000); https://doi.org/10.1016/S0378-4371(00)00071-6 DOI: https://doi.org/10.1016/S0378-4371(00)00071-6
A. Friedman, PDE problems arising in mathematical biology, Netw. Heterog. Media, 7, № 4, 691 – 703 (2012); https://doi.org/10.3934/nhm.2012.7.691. DOI: https://doi.org/10.3934/nhm.2012.7.691
E. Montefusco, B. Pellacci, G. Verzini, Fractional diffusion with Neumann boundary conditions: the logistic equation, Discrete and Contin. Dyn. Syst. Ser. B, 18, № 8, 2175 – 2202 (2013); https://doi.org/10.3934/dcdsb.2013.18.2175. DOI: https://doi.org/10.3934/dcdsb.2013.18.2175
S. D. Ejdel`man, Ya. M. Dry`n`, Neobxody`mye y` dostatochnye uslovy`ya staby`ly`zacy`y` resheny`ya zadachy` Koshy` dlya paraboly`chesky`x psevdody`fferency`al`nyx uravneny`j, Pry`bly`zhennye metody matematy`cheskogo analy`za, 60 – 69 (1974).
Ya. M. Drin`, Vy`vchennya odnogo klasu parabolichny`x psevdody`ferenczial`ny`x operatoriv u prostorax gel`derovy`x funkczij, Dop. AN URSR. Ser. A, № 1, 19 – 21 (1974).
S. D. Ejdel`man, Ya. M. Dry`n`, Postroeny`e y` y`ssledovany`e klassy`chesky`x fundamental`nyx resheny`j zadachy` Koshy` ravnomerno paraboly`chesky`x psevdody`fferency`al`nyx uravneny`j, Mat. y`ssled., vyp. 63, 60 – 69 (1981).
M. V. Fedoryuk, Asy`mptoty`ka funkcy`y` Gry`na psevdody`fferency`al`nogo paraboly`cheskogo uravneny`ya, Dy`fferencz. uravneny`ya, 14, № 7, 1296 – 1301 (1978).
W. R. Schneider, Stable distributions: Fox function representation and generalization, Lect. Notes Phys, 262, 497 – 511 (1986), https://doi.org/10.1007/3540171665_92 DOI: https://doi.org/10.1007/3540171665_92
R. M. Blumenthal, R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95, 263 – 273 (1960), https://doi.org/10.2307/1993291 DOI: https://doi.org/10.1090/S0002-9947-1960-0119247-6
A. N. Kochubej, Paraboly`chesky`e psevdody`fferency`al`nye uravneny`ya, gy`persy`ngulyarnye integraly i markovcky`e processy, Y`zv. AN SSSR. Ser. mat., 52, № 5, 909 – 934 (1988).
S. D. Eidelman, S. D. Ivasyshen, A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkh¨auser, Basel (2004), https://doi.org/10.1007/978-3-0348-7844-9 DOI: https://doi.org/10.1007/978-3-0348-7844-9
V. A. Litovchenko, Zadacha Koshi z operatorom Rissa drobovogo dy`ferencziyuvannya, Ukr. mat. zhurn., 57, № 12, 1653 – 1667 (2005).
V. A. Litovchenko, Zadacha Koshy` dlya odnogo klassa paraboly`chesky`x psevdody`fferency`al`nyx sy`stem s negladky`my` sy`mvolamy`, Sy`b. mat. zhurn, 49, № 2, 375 – 394 (2008); https://doi.org/10.1007/s11202-008-0030-z. DOI: https://doi.org/10.1007/s11202-008-0030-z
V. Knopova, A. Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by $alpha$ -stable noise, Ann. Inst. Henri Poincar´e Probab. Stat., 54, № 1, 100 – 140 (2018); https://doi.org/10.1214/16-AIHP796. DOI: https://doi.org/10.1214/16-AIHP796
V. P. Knopova, A. N. Kochubei, A. M. Kulik, Parametrix methods for equations with fractional Laplacians, vol. 2, Fractional differential equations, De Gruyter, Berlin, Boston (2019), p. 267 – 298; https://doi.org/10.1515/9783110571660-013. DOI: https://doi.org/10.1515/9783110571660-013
Wei Liu, Renming Song, Longjie Xie, Gradient estimates for the fundamental solution of Levy type operator, Adv. Nonlinear Anal., 9, № 1, 1453 – 1462 (2020); https://doi.org/10.1515/anona-2020-0062. DOI: https://doi.org/10.1515/anona-2020-0062
Y`. M. Gel`fand, G. E. Shy`lov, Prostranstva osnovnыx y` obobshhennыx funkcy`j, Fy`zmatgy`z, Moskva (1958).
L. Schwartz, Theorie Des Distributions, Hermann Paris (1951).
O. Frostman, Potentiel d’equilibre et capacit'e des ensembles avec quelques applications a la th´eorie des fonctions, Medd. Lunds Univ. Mat. S´emin, 3, 1 – 118 (1935).
M. Riesz, Integrales de Riemann – Liouville et potentiels, Acta Litt. Acad. Sci. Szeged, 9, 1 – 42 (1938).
S. L. Sobolev, Ob odnoj teoreme funkcy`onal`nogo analy`za, Mat. sb., 4, № 3, 471 – 497 (1938).
G.Thorin, Convexiti theorems, Comm. Semin. Math. Univ. Lund. Uppsala, 9, 1 – 57 (1948).
E. Stein, The characterisation of functions arising as potentials, Bull. Amer. Math. Soc., 67, № 1, 102 – 104 (1961), https://doi.org/10.1090/S0002-9904-1961-10517-X DOI: https://doi.org/10.1090/S0002-9904-1961-10517-X
P.Y. Ly`zorky`n, Opy`sany`e prostranstv $L^r_p(R^n)$ v termy`nax raznostnyx sy`ngulyarnyx y`ntegralov, Mat. sb., 81, № 1, 79 – 91 (1970).
S. G. Samko, O prostranstvax ry`ssovyx potency`alov O prostranstvax ry`ssovyx potency`alov, Y`zv. AN SSSR. Ser. mat., 40, № 5, 1143 – 1172 (1976).
Copyright (c) 2022 Владислав Антонович Літовченко
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.