scholarly journals Solvability of a noncoercive initial-boundary value problem for the Stokes system in Hölder classes of functions (the half-space case) (in Russian)

1989 ◽  
Vol 8 (4) ◽  
pp. 329-347 ◽  
Author(s):  
Ilya Mogilevskii ◽  
Vsevolod Solonnikov
Keyword(s):  
Boundary Value Problem ◽  
Half Space ◽  
Stokes System ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Hölder Classes ◽  
Space Case ◽  
Initial Boundary Value ◽  
Holder Classes

scholarly journals Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hailiang Li ◽  
Houzhi Tang ◽  
Haitao Wang
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Classical Solution ◽  
Half Space ◽  
Stokes Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
One Dimensional ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

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