The discrete Morse flow method for parabolic p -Laplacian systems
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The discrete Morse flow method for parabolic p‑Laplacian systems Nobuyuki Kato1 · Masashi Misawa2 · Yoshihiko Yamaura3 Received: 28 May 2020 / Accepted: 11 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A regularity for a parabolic p-Laplacian system (p > 2) is studied by the use of the discrete Morse flow method which is known as one of the ways to approximate a solution to parabolic partial differential equations. Our approximate solution is constructed from the sequence of minimizers of variational functionals whose Euler–Lagrange equations are the time discretized p-Laplacian system. The aim of this paper is to establish that the regularity estimates for the approximate solution hold uniformly on two approximation parameters and show strong convergence of the approximate solution. Keywords Parabolic p-Laplacian system · Time-discretization · Discrete Morse flow method · Approximation · Regularity Mathematics Subject Classification 35A05 · 35B65 · 35J50 · 35K45 · 35K65 · 39A12
Dedicated to Professor Norio Kikuchi’s 80th birthday. * Nobuyuki Kato [email protected] Masashi Misawa mmisawa@kumamoto‑u.ac.jp Yoshihiko Yamaura [email protected]‑u.ac.jp 1
Department of Mathematics, Nippon Institute of Technology, 4‑1, Gakuendai, Miyashiro, Minami‑Saitama, Saitama 345‑8501, Japan
2
Department of Mathematics, Kumamoto University, 2‑39‑1, Kurokami, Chuo, Kumamoto 860‑8555, Japan
3
Department of Mathematics, College of Humanities and Sciences, Nihon University, 3‑25‑40, Sakurajosui, Setagaya, Tokyo 156‑8550, Japan
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1 Introduction Let T be a positive number, 𝛺 a bounded domain in the m-dimensional Euclidean space ℝm ( m ∈ ℕ , m ≥ 2 ) and Q ∶= (0, T) × 𝛺 . Throughout this article, we fix p > 2 . We consider the initial-boundary value problem for the parabolic p-Laplacian system:
⎧ 𝜕 u = div��∇u�p−2 ∇u� in Q, ⎪ t in 𝛺, ⎨ u(0, ⋅) = u0 ⎪ u(t, ⋅) = u0 on 𝜕𝛺. ⎩
(1.1)
Here, the unknown function u is a vector-valued function with values into ℝM ( M ∈ ℕ , M ≥ 1 ), defined on Q, and the initial datum u0 is assumed to belong to the Sobolev space W 1,p (𝛺, ℝM ) . The parabolic p-Laplacian system (1.1) is the typical degenerate and singular parabolic system. The existence and regularity of a weak solution have already been established in [2–4], and their results are the fundamental theorems for the degenerate and singular parabolic equations and systems. We shall consider how to approximate a weak solution to (1.1) and aim at studying how to get some uniform regularity estimates for the approximating solutions. The approximation method considered here is named the discrete Morse flow (abbreviated to DMF) method. Let us explain the DMF method for (1.1): We 1,p denote by Wu0 (𝛺, ℝM ) the set of functions in the usual Sobolev space W 1,p (𝛺, ℝM ) , whose trace on the boundary 𝜕𝛺 is u0 . Set h = T∕N for a positive number N, and tn ∶= nh for n = 0, 1, … , N . Starting from the initial datum
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