リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「FRACTIONAL TIME DIFFERENTIAL EQUATIONS AS A SINGULAR LIMIT OF THE KOBAYASHI-WARREN-CARTER SYSTEM」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

FRACTIONAL TIME DIFFERENTIAL EQUATIONS AS A SINGULAR LIMIT OF THE KOBAYASHI-WARREN-CARTER SYSTEM

GIGA, YOSHIKAZU KUBO, AYATO KURODA, HIROTOSHI OKAMOTO, JUN SAKAKIBARA, KOYA UESAKA, MASAAKI 北海道大学

2023.06.29

概要

We consider the Kobayashi–Warren–Carter system, introduced in [11, 12, 13], to
model evolutions of structures in a multi-grain problem. It is a kind of phase-field
system in a domain Ω in Rn , formally a gradient flow of the energy
Z
ε
ε
(1.1)
EKWC
(u, v) :=
α0 (v)|∇u| + EsMM
(v),

Z
Z
ε
1
ε
EsMM
(v) :=
(1.2)
|∇v|2 dx +
F (v) dx.
Ω 2
Ω 2ε
Here, α0 (v) ≥ 0 is a given function, typically α0 (v) = sv 2 , with a constant s > 0,
and F (v) is a single-well potential, typically F (v) = a2 (v − 1)2 , with a constant
ε
is often called a single-well Modica–Mortola functional.
a > 0. ...

この論文で使われている画像

参考文献

[1] F. Andreu-Vaillo, V. Caselles, J. M, Maz´

on, Parabolic quasilinear equations minimizing linear growth functionals. Progress in Mathematics, 223. Birkh¨

auser Verlag, Basel,

2004. xiv+340 pp.

[2] H. Br´

ezis, Op´

erateurs maximaux monotones et semi-groupes de contractions dans les

espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matem´

atica,

No. 50. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp.

[3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian.

Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260.

[4] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations. Taniguchi

Conference on Mathematics Nara ’98, 93–125, Adv. Stud. Pure Math., 31, Math. Soc.

Japan, Tokyo, 2001.

[5] M.-H. Giga, Y. Giga and J. Saal, Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions. Progress in Nonlinear Differential

A SINGULAR LIMIT OF THE KOBAYASHI-WARREN-CARTER SYSTEM

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

23

Equations and their Applications, 79. Birkh¨

auser Boston, Ltd., Boston, MA, 2010.

xviii+294 pp.

Y. Giga, H. Mitake and S. Sato, On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation. J. Differential Equations

316 (2022), 364–386.

Y. Giga and T. Namba, Well-posedness of Hamilton–Jacobi equations with Caputo’s

time fractional derivative. Comm. Partial Differential Equations 42 (2017), no. 7,

1088–1120.

Y. Giga, J. Okamoto, K. Sakakibara and M. Uesaka, On a singular limit of the

Kobayashi–Warren–Carter energy. Indiana Univ. Math. J., to appear.

Y. Giga, J. Okamoto and M. Uesaka, A finer singular limit of a single-well Modica–

Mortola functional and its applications to the Kobayashi–Warren–Carter energy. Adv.

Calc. Var. 16 (2023), 163–182.

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion.

Appl. Math. 53 (2008), no. 5, 433–454.

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundaries.

Phys. D 140 (2000), 141–150.

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular

diffusivity. Free boundary problems: theory and applications, II (Chiba, 1999), 283–

294, GAKUTO Internat. Ser. Math. Sci. Appl. 14, Gakk¯

otosho, Tokyo, 2000.

R. Kobayashi, J. A. Warren and W. C. Carter, Modeling grain boundaries using a

phase-field technique. J. Cryst. Growth 211 (2000), 18–20.

Y. K¯

omura, Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967),

493–507.

A. Kubica, K. Ryszewska and M. Yamamoto, Time-fractional differential equations –

a theoretical introduction. SpringerBriefs in Mathematics. Springer, Singapore, 2020.

x+134 pp.

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. [2013

reprint of the 1995 original] Modern Birkh¨

auser Classics. Birkh¨

auser/Springer Basel

AG, Basel, 1995. xviii+424 pp.

S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi–Warren–Carter

system. Calc. Var. Partial Differential Equations 51 (2014), no. 3–4, 621–656.

S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the

Kobayashi–Warren–Carter system. Nonlinearity 30 (2017), no. 7, 2752–2784.

S. Moll, K. Shirakawa and H. Watanabe, Kobayashi–Warren–Carter type systems with

nonhomogeneous Dirichlet boundary data for crystalline orientation, in preparation.

T. Namba, On existence and uniqueness of viscosity solutions for second order fully

nonlinear PDEs with Caputo time fractional derivatives. NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 3, Paper No. 23, 39 pp.

I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their

applications. Mathematics in Science and Engineering, 198. Academic Press, Inc., San

Diego, CA, 1999. xxiv+340 pp.

M. H. Protter and H. F. Weinberger, Maximum principles in differential equations.

Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. x+261 pp.

K, Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional

diffusion-wave equations and applications to some inverse problems. J. Math. Anal.

Appl. 382 (2011), no. 1, 426–447.

K. Shirakawa and H. Watanabe, Energy-dissipative solution to a one-dimensional

phase field model of grain boundary motion. Discrete Contin. Dyn. Syst. Ser. S 7

(2014), no. 1, 139–159.

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase

field systems associated with grain boundary motion. Math. Ann. 356 (2013), no. 1,

301–330.

E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo

time derivative. J. Differential Equations 262 (2017), no. 12, 6018–6046.

H. Watanabe and K. Shirakawa, Qualitative properties of a one-dimensional phase-field

system associated with grain boundary, in: Nonlinear Analysis in Interdisciplinary

24

Y. GIGA, A. KUBO, H. KURODA, J. OKAMOTO, K. SAKAKIBARA, AND M. UESAKA

Sciences – Modellings, Theory and Simulations, GAKUTO Internat. Ser. Math. Sci.

Appl. 36, Gakk¯

otosho, Tokyo (2013), 301–328.

[28] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in

Hilbert spaces. Funkcial. Ekvac. 52 (2009), no. 1, 1–18.

[29] R. Zacher, Time fractional diffusion equations: solution concepts, regularity, and longtime behavior. Handbook of fractional calculus with applications. Vol. 2, 159–179, De

Gruyter, Berlin, 2019.

(Y. Giga) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1

Komaba, Meguro-ku, Tokyo 153-8914, Japan.

Email address: labgiga@ms.u-tokyo.ac.jp.

(A. Kubo) Department of Mathematics, Faculty of Science, Hokkaido University,

Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan.

Email address: kubo.ayato.j8@elms.hokudai.ac.jp

(H. Kuroda) Department of Mathematics, Faculty of Science, Hokkaido University,

Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan.

Email address: kuro@math.sci.hokudai.ac.jp

(J. Okamoto) Institute for the Advanced Study of Human Biology, Kyoto University,

Yoshida-Konoe-Cho, Sakyo-ku, Kyoto 606-8501, Japan.

Email address: okamoto.jun.8n@kyoto-u.ac.jp

(K. Sakakibara) Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan;

RIKEN iTHEMS, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan.

Email address: ksakaki@se.kanazawa-u.ac.jp

(M. Uesaka) Graduate School of Mathematical Sciences, The University of Tokyo,

3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan; Arithmer Inc., ONEST Hongo Square

3F, 1-24-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

...

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文

一発検索!

この論文の関連論文を見る