| Project/Area Number |
09440056
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B).
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
解析学
|
|---|
| Research Institution | Tohoku University (1999-2000)
Nagoya University (1997-1998) |
|---|
Principal Investigator |
KOZONO Hideo Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (00195728)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
NAGASAWA Takeyuki Tohoku University, Graduate School of Science, Ass.Prof., 大学院・理学研究科, 助教授 (70202223)
TSUTSUMI Yoshio Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (10180027)
TAKAGI Izumi Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (40154744)
TACHIZAWA Kazuya Tohoku University, Graduate School of Science, Lect., 大学院・理学研究科, 講師 (80227090)
CHIHARA Hiroyuki Tohoku University, Graduate School of Science, Ass.Prof., 大学院・理学研究科, 助教授 (70273068)
石毛 和弘 名古屋大学, 大学院・多元数理科学研究科, 助教授 (90272020)
三宅 正武 名古屋大学, 大学院・多元数理科学研究科, 教授 (70019496)
小川 卓克 名古屋大学, 大学院・多元数理科学研究科, 助教授 (20224107)
中村 周 東京大学, 大学院・数理科学研究科, 教授 (50183520)
加藤 義夫 名古屋大学, 大学院・多元数理科学研究科, 教授 (70023968)
|
|---|
| Project Period (FY) |
1997 – 2000
|
| Project Status |
Completed (Fiscal Year 2000)
|
| Budget Amount *help |
¥13,600,000 (Direct Cost: ¥13,600,000)
Fiscal Year 2000: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1998: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1997: ¥3,800,000 (Direct Cost: ¥3,800,000)
|
|---|
| Keywords | Navier-Stokes Equations / Sobolur Space / Inter polation space / Fowwer transform / singular integral operation / Stoker operator / Lorentz space / exterior problem / ストークス方程式 / 応力テンソル / エネルギー不等式 / 安定性 / オイラー方程式 / 双線形作用素 / ハーディ空間 / BMO関数 / ベゾフ空間 / ソボレフ不等式 / 爆発解 / 斉次ソボレク空間 / L^r-空間 / 弱解の一意性 / 弱解の正則性 / 半線形放物形方程式 / 除去可能と特異点 |
|---|
| Research Abstract |
In a domain Ω⊂R^n, consider a weak solution u of the Navier-Stokes equations in the class u∈L^∞ (0, T ; L^n (Ω)). If lim sup_<t-t_*-0>‖u (t) ‖^n_n-‖u (t_*) ‖^n_n is small at each point of t_*∈ (0, T), then u is regular on Ω^^-× (0, T). As an application, we give a precise characterization of the singular time, i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_*<T, then either lim sup_<t-T_*-0>‖u (t) ‖_<L^n (Ω) >= +∞, or u (t) oscillates in L^n (Ω) around the weak limit w-lim_<t-T_*-0>u (t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in L^n (Ω) becomes regular.
Consider the nonstationary Navier-Stokes equations in Ω× (0, T), where Ω is a domain in R^3. We show that there is an absolute constant ε_0 such that every weak solution u with the property sup_<t∈ (a, b) >‖u (t) ‖^3_W (D) 【less than or equal】ε_0 is necessarily of class C^∞ in the space-time variables on any compact subset of D× (a, b), where D ⊂⊂Ω and 0<a<b<T.As an application, we prove that if the weak solution u behaves around (x_0, t_0) ∈Ω× (0, T) like u (x, t) =o (|x-x_0|^<-1>) as x→x_0 uniformly in t in some neighborhood of t_0, then (x_0, t_0) is a removable singularity of u.
Consider weak solutions w of the Navier-Stokes equations in Serrin's class
w∈L^α (0, ∞ ; L^q (Ω)) for 2/α + 3/q = 1 with 3<q【less than or equal】∞,
where Ω is a general unbounded domain in R^3. We shall show that although the inital and exteral disturbances from w are large, every perturbed flow u with the energy inequality converges asymptotically to w as
‖υ (t) -w (t) ‖_<L^2 (Ω) >→0, ‖▽υ(t) -▽w (t) ‖_<L^2 (Ω) >=O (t^<-1/2>) as t→∞.
|
