Asymptotic behavior of solutions and stability of nonlinear waves for equations of gas motion

气体运动方程解的渐近行为和非线性波的稳定性

• 批准号: 14340047
• 负责人: KAWASHIMA Shuichi
• 金额: $ 7.3万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340047/
• 关键词: Viscous conservation laws; Hyperbolic conservation laws with relaxation; Compressible Navier-Stokes equation; Dissipative Timoshenko system; Dissipative wave equations; Time global solutions; Asymptotic stability; Time decay estimates; 緩和的双曲系; 双曲

We studied asymptotic behavior of solutions and stability of nonlinear waves for equations of gas motion with dissipative structure.1.We developed the energy method in the Sobolev space W^{1,p} for n-dimensional scalar viscous conservation law and derived the optimal decay estimates in W^{1,p}. The method was also applied to the stability problem for rarefaction waves and stationary waves.2.We introduced the notion of entropy for n-dimensional hyperbolic conservation laws with relaxation and developed the Chapman-Enskog theory. Moreover, we proved the global existence and optimal decay of solutions in a L^2 type Sobolev space.3.For the compressible Navier-Stokes equation in the n-dimensional half space, we proved the asymptotic stability of planar stationary waves. To develop the theory in the Sobolev space of order [n/2]+1, we need additional considerations for local existence results.4.For the dissipative Timoshenko system, we derived qualitative decay estimates of solutions by applying the energy method in Fourier space. We found that the dissipative structure is so weak in high frequency region and it causes the regularity loss in the decay estimates.5.For dissipative wave equation with a nonlinear convection term, we proved the global existence and optimal decay of solutions in L^p. Moreover, we showed that the solution approaches the nonlinear diffusion waves given in terms of the self similar solutions of the Burgers equation. Derivation of detailed pointwise estimates of the fundamental solutions is crucial in the proof.

研究了具有耗散结构的气体运动方程解的渐近性和非线性波的稳定性。我们在Sobolev空间W^{1，p}中建立了n维标量粘性守恒律的能量法，并推导了W^{1，p}中的最优衰减估计。该方法也适用于稀疏波和驻波的稳定性问题。我们引入了n维松弛双曲守恒定律的熵的概念，并发展了Chapman-Enskog理论。此外，我们还证明了L^2型Sobolev空间解的全局存在性和最优衰减性。对于n维半空间中的可压缩Navier-Stokes方程，证明了平面驻波的渐近稳定性。为了在[n/2]+1阶的Sobolev空间中发展理论，我们需要额外考虑局部存在性结果。对于耗散的Timoshenko系统，我们在傅里叶空间中应用能量法推导了解的定性衰减估计。我们发现在高频区域耗散结构很弱，导致衰减估计的规律性丧失。对于具有非线性对流项的耗散波动方程，证明了其解在L^p上的全局存在性和最优衰减性。此外，我们还证明了解接近于用Burgers方程的自相似解给出的非线性扩散波。对基本解的详细的逐点估计的推导在证明中是至关重要的。

项目成果

A limit problem in natural convection

自然对流的极限问题

• 期刊: Nonlinear Differential Equations and Applications (To appear)(印刷中)
• 作者: Yoshiyuki Kagei;M.Ruzicka;G.Thaeter
• 通讯作者: G.Thaeter

Decay property of regularity-loss type and application to some hyperbolic-elliptic systems

正则性损失型的衰变性质及其在某些双曲椭圆系统中的应用

• 发表时间: 2006
• 期刊: Math. Models Meth. Appl. Sci. 16
• 作者: 早川岳人;他;T.Hosono
• 通讯作者: T.Hosono

L^p-L^q type estimate for semi-linear dumped wave equation in two dimensions

二维半线性倾倒波动方程的L^p-L^q型估计

• 发表时间: 2004
• 期刊: J. Differential Equations 203
• 作者: Y.Kagei;Y.Ohtsubo;Y.Goto;T.Kobayashi;Y.Ohtsubo;S.Kawashima;藤田敏治;S.Kawashima;T.Fujita;S.Kawashima;H.Hyakutake;K.Ohgane;H.Hyakutake;T.Nakamura;H.Kawasaki;T.Ogawa;S.Iwamoto;T.Ogawa;Y.Ohtsubo;H.Usami;T.Hosono
• 通讯作者: T.Hosono

S.Kawashima: "Asymptotic stability of the stationary solution to compressible Navier-Stokes equations in the half-space"Commun.Math.Phys.. 240. 483-500 (2003)

S.Kawashima：“半空间中可压缩纳维-斯托克斯方程的平稳解的渐近稳定性”Commun.Math.Phys.. 240. 483-500 (2003)

T.Ogawa: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain"J.Math.Fluid Mech.. 5. 17-23 (2003)

T.Okawa：“关于有界域中 3-D 欧拉方程的爆炸准则的注释”J.Math.Fluid Mech.. 5. 17-23 (2003)