An application of real aualytical wethad to nonlinear evolution eguations

实数分析湿法在非线性演化方程中的应用

• 批准号: 09440047
• 负责人: SHIBATA Yoshihiro
• 金额: $ 8.77万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440047/
• 关键词: 3 dim. Exterior Domain; Navier-Stokes Equation; Physically Reasonable Solution; Stability; 2 dim. Exterior Domain; LィイD2pィエD2-LィイD2qィエD2 estimate; Stokes semigroup; Oseen semigroup; 非圧縮性粘性流体; 圧縮性粘性流体; L_<p,∞>空間; diffusion wave; 2相問題; Ginzburg

We studied the initial-boundary value problem for Navier-Stokes equation in 3 dimensional exterior domain which describes the motion of incompressible viscous fluids. In 1930, J. Leray proved the existence of its weak solutions without uniqueness. But, we can not get any qualitative property of the motion of fluid from Leray's solutions. In 1950, R. Finn started to study the qualitative property of soltuions to the stationary Navier-Stokes equation, which was termed physically reasonable solutions (prs) by him. He proved the unique existence theorem of prs for small external force and the velocity uィイD2∞ィエD2 at infinity. After Finn, Heywood proved the stability of prs in the LィイD22ィエD2 framework, which now become one of the most important base of numerical investigation of the fluid motion. But, prs does not belong to LィイD22ィエD2 space, and therefore we have to study the stability of prs in the class to which prs belongs. This problem remained more than 30 years. In our present study, w … More e solved this problem. We used the following argument. We took the Oseen approcimation in the 3 dim. Exterior domain, and then we investigated the local energy estimate near the boundary of optimal order of the Oseen equation by showing the fractional differentiablity of Oseen resolvent near the origin. Combinig this and LィイD2pィエD2-LィイD2qィエD2 estimate in the whole space by cut-off technique, we proved the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of the Oseen semigroup in 3 dim. Exterior domain. The most important point is that all the constant appearing in the estimate is independent of uィイD2∞ィエD2. By using this estimate, we could solve the stability problem of Finn's prs.(2) In 2 dimensinal case, we know the unique existence of Leray's weak solutions. But, since the Stokes fundamental solution has logarihmic singularity, we know less property of solution to NS in the exterior domain or even the whole space compared with 3 dim. Case. We could obtain the asymptotic expansion of Stokes resolvent near the origin and we found that the logarithmic singularity is canceled out by the reflection phenomenon near the bounday, and therefore we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate (1< q≦ p≦ ∞) of Stokes semigroup in a 2 dim. Exterior domain by using the similar argument to the 3 dim. Case. Applying this estimate, we could show the best convergent rate of weak solutions to NS when time goes to infinity.(3) We considered the motion of the compressible viscous fluid too. Extending the method developed in study (1), we obtained the optimal LィイD2pィエD2-LィイD2qィエD2 estimate of solutions to linearized eqautions in the 3 dim. Exterior domain, which is applied to obtain the optimal convergent rate of solutions to the original nonlinear probolem at time infinity. Less

研究了三维不可压缩粘性流体运动的Navier-Stokes方程的初边值问题。1930年，J.Leray证明了其弱解的存在性，但不存在唯一性。但是，我们不能从Leray解中得到流体运动的任何定性性质。1950年，R.Finn开始研究定常的Navier-Stokes方程的解的定性性质，他称之为物理合理解。证明了当外力较小时，速度为u，ィイ，d2，∞ィエ，d2，无穷大时的解存在唯一性定理。继Finn之后，Heywood证明了在LィイD22ィエD2框架下的数值解的稳定性，该框架现已成为流体运动数值研究的重要基础之一。但是，伪随机序列不属于LィイD22ィエD2空间，因此我们必须在伪随机序列所属的类中研究其稳定性。这个问题持续了30多年。在我们目前的研究中，w…更多的人解决了这个问题。我们使用了以下论点。我们在3DIME中拍摄了OSee APPRIMATION。然后，通过证明Oseen预解在原点附近的分数次可微性，研究了Oseen方程最优阶边界附近的局部能量估计。利用截断技巧将其与全空间上的LィイD2pィエD2-LィイD2qィエD2估计相结合，证明了三维Oseen半群的最优LィイD2pィエD2-LィイD2qィエD2估计.外部域。最重要的一点是，估计中出现的所有常数都与uィイD2，∞ィエD2无关。在2维情形下，我们知道了Leray弱解的唯一存在性。但是，由于Stokes基本解具有对数奇性，与三维相比，我们对NS的解在外部区域甚至整个空间的性质知之甚少。凯斯。我们得到了Stokes预解在原点附近的渐近展开式，并发现对数奇性被边界附近的反射现象所抵消，从而得到了二维Stokes半群的最优LィイD2pィエD2-LィイD2qィエD2估计(1&lt；q≦p≦∞)。通过使用类似于3维的论据来确定外部区域。凯斯。应用这个估计，我们可以给出当时间变到无穷大时NS弱解的最佳收敛速度。(3)我们还考虑了可压缩粘性流体的运动。推广了文献[1]的方法，得到了三维线性化方程的最优解的LィイD2pィエD2-LィイD2qィエD2估计。外部域，用于获得原非线性问题解在时间无穷远的最优收敛速度。较少

项目成果

W.Dan & Y.Shibata: "On the Lg-L∞ estimate of the stokes semigroup in a two dimensional exterior domain"Pacific J.Math.. 189・2. 223-239 (1999)

W.Dan & Y.Shibata：“关于二维外部域中斯托克斯半群的 Lg-L∞ 估计”Pacific J.Math.. 189・2. 223-239 (1999)

W.Dan & Y.Shibata: "On the L_q-L_r estimate of Mu stokes semigroup in a two dimainonal exterion domain" J.Math.Soc.Japan. 181-207 (1999)

丹

Y. Shibata & W. Dan: "On the L_q-L_v estimate of the stokes luigroup in a two dimensional exbvior dowa in" J. Math. Soc. Japan. (発表予定). (1998)

Y. Shibata 和 W. Dan：“关于二维 exbvior dowa 中的 L_q-L_v 估计”，《J. Math》（即将出版）。

Y. Shibata: "An extericer inifical bowdary value problem for the Navier-Stokes equation" Mathematical Sciences and Applications. 10. 431-446 (1997)

Y. Shibata：“纳维-斯托克斯方程的一个外部 inifical Bowdary 值问题”《数学科学与应用》。

T.KOBAYASHI & W.ZATACZKOWSKL: "On global motion of a compressible viscous fuvid with boundary slip condition"Applications Mathematical. 26. 159-194 (1999)

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