Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
An Introduction to Maximal Regularity for Parabolic Evolution Equations Kapitel lesen Erstes Kapitel lesen

2021 | OriginalPaper | Buchkapitel

Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

verfasst von : Takayoshi Ogawa

Erschienen in: Nonlinear Partial Differential Equations for Future Applications

Verlag: Springer Singapore

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We consider singular limit problems of the Cauchy problem for the Patlak-Keller-Segel equation and related problems appeared in the theory of medical and biochemical dynamics. It is shown that the solution to the Patlak-Keller-Segel equation in a scaling critical function class converges strongly to a solution of the drift-diffusion system of parabolic-elliptic equations as the relaxation time parameter \(\tau \rightarrow \infty \). Analogous problem related to the Chaplain-Anderson model for cancer growth model is also presented as well as Arzhimer’s model that involves the multi-component drift-diffusion system. For the proof, we use generalized maximal regularity for the heat equations and systematically apply embeddings between the interpolation spaces shown in [40, 41]. The argument requires generalized version of maximal regularity developed in [40, 61], for the Cauchy problem of the heat equation.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Uniform Regularity for a Compressible Gross-Pitaevskii-Navier-Stokes System
Nächstes Kapitel HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control
Fußnoten
1
It is also valid for \(n=2\). Assuming further \(\lambda >0\) and \(u_0\in \dot{B}^0_{1,4}(\mathbb R^n)\).
 
2
The choice of T is independent of \(\tau >1\).
 
3
The choice of T is independent of \(\tau >1\).
 
Literatur
1.
Amann,  H.: Linear and Quasilinear Parabolic Problems. Vol. I Abstract Linear Theory. Monographs in Mathematics, vol. 89. Birkhäuser Verlag, Basel, Boston, Berlin (1995)
2.
Bedrossian, J.: Large mass global solutions for a class of \(L^1\)-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel model. Commun. P.D.E. 40, 1119–1136 (2015)
3.
Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48, 356–365 (1962)MathSciNetMATHCrossRef
4.
Biler, P.: Existence and nonexistence of solutions for a model of gravitational interaction of particles, III. Colloq. Math. 68, 229–239 (1995)MathSciNetMATHCrossRef
5.
Biler, P., Brandolese, L.: On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modeling chemotaxis. Studia Math. 193, 241–261 (2009)MathSciNetMATHCrossRef
6.
Biler, P., Nadzieja, T.: Existence and nonexistence of solutions for a model of gravitational interactions of particles I. Colloq. Math. 66, 319–334 (1994)MathSciNetMATHCrossRef
7.
Biler, P., Nadzieja, T.: A nonlocal singular parabolic problem modeling gravitational interaction of particles. Adv. Differ. Eq. 3, 177–197 (1998)MATH
8.
Brezis, H.: Anal. Fonctionnelle. Théorie et applications, Masson, Paris (1983)
9.
Bournaveas, N., Calvez, V.: The one-dimensional Keller-Segel model with fractional diffusion of cells. Nonlinearity 23, 923–935 (2010)MathSciNetMATHCrossRef
10.
Calvez, V., Corrias, L., Ebde, M.: Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension. Commun. Partial Differ. Eq. 37, 561–584 (2012)MathSciNetMATHCrossRef
11.
Chandrasekhar, S.: An Introduction to the Study of Stellar Structure, p. 509. Dover Publications Inc., New York (1957)
12.
Chaplain, M.A.J., Anderson, A.R.A.: Mathematical Modeling of Tissue Invasion, “Cancer Modeling and Simulation”, pp. 269–297. Math. Biol. Med. Ser. Chapman & Hall/CRC, Boca Raton, FL (2003)
13.
14.
Corrias, L., Escobedo, M., Matos, J.: Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane. J. Differ. Eq. 257, 1840–1878 (2014)MathSciNetMATHCrossRef
15.
Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis system in height space dimensions. Milan J. Math. 72, 1–28 (2004)MathSciNetMATHCrossRef
16.
17.
Danchin, R.: Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Commun. Partial Differ. Eq. 32, 1373–1397 (2007)MathSciNetMATHCrossRef
18.
Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pure Appl. 54, 305–387 (1975)MATH
19.
Denk, R., Hieber, M., Prüss, J.: \(\cal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. AMS 166(788) (2003)
20.
21.
Fujie, K., Ito, A., Yokota, T.: Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type. Adv. Math. Sci. Appl. 24, 67–84 (2014)MathSciNetMATH
22.
Fujie, K., Ito, A., Winkler, M., Yokota, T.: Stabilization in a chemotaxis model for tumor invasion. Discrete Contin. Dyn. Syst. Ser. A 36, 151–169 (2016)MathSciNetMATH
23.
Fujie, K., Senba, T.: Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension. J. Differ. Eq. 266, 942–976 (2019)MathSciNetMATHCrossRef
24.
Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)
25.
26.
Hieber, M., Prüss, J.: Heat kernels and maximal \(L^p\)-\(L^q\) estimates for parabolic evolution equations. Commun. P.D.E. 22, 1674–1669 (1997)
27.
28.
Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modeling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MATHCrossRef
29.
Jin, H.Y., Liu, Z.: Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space. Appl. Math. Lett. 47, 13–20 (2015)MathSciNetMATHCrossRef
30.
Jüngel, A.: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Model. Meth. Appl. Sci. 5, 497–518 (1995)MathSciNetMATHCrossRef
31.
32.
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetMATHCrossRef
33.
34.
Kozono, H., Sugiyama, Y., Yahagi, Y.: Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system. J. Differ. Eq. 253, 2295–2313 (2012)MathSciNetMATHCrossRef
35.
Kubo, H., Ogawa, T., Suguro, T.: Beckner type of the logarithmic Sobolev and a new type of Shannon’s inequalities and an application to the uncertainty principle. Proc. Am. Math. Soc. 147(4), 1511–1518 (2019). https://​doi.​org/​10.​1090/​proc/​14350
36.
Kurokiba, M., Nagai, T., Ogawa, T.: The uniform boundedness of the radial solution for drift-diffusion system. Commun. Pure Appl. Anal. 5, 97–106 (2006)MathSciNetMATHCrossRef
37.
Kurokiba, M., Ogawa, T.: Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type. Differ. Integral Eq. 16, 427–452 (2003)MathSciNetMATH
38.
Kurokiba, M., Ogawa, T.: \(L^p\) well-posedness of the for the drift-diffusion system arising from the semiconductor device simulation. J. Math. Anal. Appl. 342, 1052–1067 (2008)MathSciNetMATHCrossRef
39.
Kurokiba, M., Ogawa, T.: Finite time blow up for a solution to system of the drift-diffusion equations in higher dimensions. Math. Z. 284(1–2), 231–253 (2016)MathSciNetMATHCrossRef
40.
Kurokiba, M., Ogawa, T.: Singular limit problem for Keller-Segel system and drift-diffusion system in scaling critical spaces. J. Evol. Eq. 20, 421–457 (2020)MathSciNetMATHCrossRef
41.
Kurokiba, M., Ogawa, T.: Singular limit problem for the two-dimensional Keller-Segel system in scaling critical space. J. Differ. Eq. 269, 8959–8997 (2020)MathSciNetMATHCrossRef
42.
Kurokiba, M., Ogawa, T.: Singular limit problem for the generalized Keller-Segel system with the fractional diffusion in scaling critical spaces (in preparation)
43.
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. Transl. Math. Monographs, Providence, R.I. (1968)
44.
Lemarié-Rieusset, P.G.: Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space. Adv. Differ. Eq. 18, 1189–1208 (2013)MathSciNetMATH
45.
Mizoguchi, N.: Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane. Calc. Var. Partial Differ. Eq. 48, 491–505 (2013)MathSciNetMATHCrossRef
46.
Mock, M.S.: An initial value problem from semiconductor device theory, SIAM. J. Math. 5, 597–612 (1974)MathSciNetMATH
47.
Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)MathSciNetMATH
48.
Nagai, T.: Blowup of non-radial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)MathSciNetMATH
49.
Nagai, T., Ogawa, T.: Brezis-Merle inequalities and application to the global existence of the Keller-Segel equations. Commun. Contemp. Math. 13(5), 795–812 (2011)MathSciNetMATHCrossRef
50.
Nagai, T., Ogawa, T.: Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in \(\mathbb{R}^2\). Funkcial. Ekvac. 59(2), 67–112 (2016)
51.
Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima J. Math. 30, 463–497 (2000)MathSciNetMATHCrossRef
52.
Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)MathSciNetMATH
53.
Nagai, T., Yamada, T.: Global existence of solutions to the Cauchy problem for an attraction-repulsion chemotaxis system in \(\mathbb{R}^2\) in the attractive dominant case. J. Math. Anal. Appl. 462, 1519–1535 (2018)
54.
Nagai, T., Yamada, T.: Global existence of solutions to a two-dimensional attraction-repulsion chemotaxis system in the attractive dominant case with critical mass. Nonlinear Anal. (to appear)
55.
Nagai, T., Yamada, T.: Boundedness of Solutions to the Cauchy Problem for an Attraction-Repulsion Chemotaxis System in Two-Dimensional Space (preprint)
56.
57.
Ogawa, T., Shimizu, S.: End-point maximal regularity and well posedness of the two dimensional Keller-Segel system in a critical Besov space. Math. Z. 264, 601–628 (2010)MathSciNetMATHCrossRef
58.
Ogawa, T., Shimizu, S.: End-point maximal \(L^1\)-regularity for the Cauchy problem to a parabolic equation with variable coefficients. Math. Ann. 365(1), 661–705 (2016)MathSciNetMATHCrossRef
59.
Ogawa, T., Shimizu, S.: Maximal regularity for the Cauchy problem of the heat equation in BMO, Math. Nachari, (2021) to appear
60.
Ogawa, T., Wakui, H.: Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions. Anal. Appl. 14, 145–183 (2016)MathSciNetMATHCrossRef
61.
Ogawa, T., Yamane, Y.: Local well-posedness for the Cauchy problem to nonlinear heat equations of Fujita type in nearly critical Besov space. In: Springer Proceedings in Mathematical Statistics. Mathematics for Nonlinear Phenomenon-Analysis and Computation, vol. 215, pp. 215–240. Springer (2017)
62.
63.
Raczyński, A.: Stability property of the two-dimensional Keller-Segel model. Asymptot. Anal. 61, 35–59 (2009)MathSciNetMATH
64.
Senba, T.: Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domain. Funkcilaj Ekvac. 48, 247–271 (2005)MATHCrossRef
65.
Senba, T.: Blowup in infinite time of radial solutions to parabolic-elliptic system in high-dimensional Euclidean spaces. Nonlinear Anal. 70, 2549–2562 (2009)MathSciNetMATHCrossRef
66.
Shi, R., Wang, W.: Well-posedness for a model derived from an attraction-repulsion chemotaxis system. J. Math. Anal. Appl. 423, 497–520 (2015)MathSciNetMATHCrossRef
67.
Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 255–269 (1959)MathSciNetMATHCrossRef
68.
Stein, E.: Harmonic Analysis. Prinston Univ. Press, Prinston, New Jersey (1990)
69.
Sugiyama, Y., Yamamoto, M., Kato, K.: Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space. J. Differ. Eq. 258, 2983–3010 (2015)MathSciNetMATHCrossRef
70.
71.
72.
73.
Wolansky, G.: On the evolution of self-interacting clusters and applications to semi-linear equations with exponential nonlinearity. J. Anal. Math. 59, 251–272 (1992)MathSciNetMATHCrossRef
Metadaten
Titel
Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity
verfasst von
Takayoshi Ogawa
Copyright-Jahr
2021
Verlag
Springer Singapore
Buch
Nonlinear Partial Differential Equations for Future Applications

Print ISBN: 978-981-334-821-9

Electronic ISBN: 978-981-334-822-6

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-981-33-4822-6_4