Control for well-posedness about a class of non-Newtonian incompressible porous medium fluid equations

Considering the non-Newtonian fluid equation of incompressible porous media, using the properties of operator semigroup and measure space and the principle of squeezed image, Fourier analysis and a priori estimate in the measurement space are used to discuss the non-compressible porous media, the properness of the solution of the equation, its gradual behavior and its topological properties. Through the diffusion regularization method and the compressed limit compact method, we study the overall decay rate of the solution of the equation in a certain space when the initial value is sufficient. The decay estimation of the solution of the incompressible seepage equation is obtained, and the asymptotic behavior of the solution is obtained by using the double regularization model and the Duhamel principle.


Introduction
Porous media the fluid mechanics branch is a discipline with rock mechanics, porous media physics, surface physics, physical chemistry, and thermodynamics and so on. Therefore, the situation is more complex, mainly in: porous media pore area larger, viscous effect is obvious, because the flow of fluid in porous media by the pressure, so its compression cannot be ignored. Due to the role of capillary phenomenon, the role of molecular force is also more complex in our study. Thus, porous media fluids are often accompanied by complex physical and chemical processes [1][2][3][4]. Porous media Fluid mechanics is widely used in water conservancy, marine, petroleum, geothermal, soil, environment, aviation, aerospace and other fields.
Using the Darcy rule, we can establish the fluid equation model of the porous medium as

Preliminaries
We focus on the flow of a class of incompressible non-Newtonian fluids in porous media, and find out the suitability and gradual behavior of a class of non-Newtonian fluids in porous media [9][10]. When a non-Newtonian fluid flows in a porous medium, its trajectory function is as follows: is the position of a particle of the fluid at time and its initial position is is the region of the initial region N R   arrives after time .

Main Conclusion
We consider the following mathematical model for a class of incompressible non-Newtonian porous media problems:

T x t and  
, f x t are the real functions about and , [0, ) p

 
We study the suitability of the equation and the asymptotic behavior of the solution. Note that the dissipation mechanism is given by the fractional Laplacian operator and that the velocity is related to the pressure, and we will use the new tools and methods to get the following conclusions: Theorem suppose

Proof of Conclusion
In order to give the main conclusion of the proof, we first introduce the following lemma [11] Lemma 1 suppose : is a bounded linear operator for Banach space , there exists a constant , for any 1 2 , , , n      ,there holds [7]   Have unique solutions , Proof: use compressed image principle, for small positive number , construct an inspection For any E   , Notice that   On the other hand, by the nature of the film, the direct calculation is available: On the other hand，From direct calculation, one can get Let , we come to the conclusion.

Simulation and Numerical Experiments
In order to verify the correctness of the theoretical analysis of the above equation, we simulate and simulate the experiment. We use COMSOL software to simulate the parameter setting of the above equation and the analysis of the results. It can be shown that grid 1 divides region along the X-axis direction and Y-axis direction (n is a positive integer) to form a parallel linear triangular mesh. Mesh 2 is a parallel linear triangular mesh formed by dividing the region by an N-division in the X-axis direction and dividing the M-divisions in the Y-axis direction with a mesh ratio of / 8 m n  As can be seen from the data in the table, the error order of grid 1 and grid 2 is the same, and its large time behavior is stable. To achieve the optimal order and convergence effect, which is consistent with the previous theoretical analysis of the results

Conclusion
In this paper, we study the global fitness of the solution of the fluid equation in the porous medium with nonlinear boundary source nonlinear diffusion of incompressible fluid. Based on the non-Newton fluid equation of the incompressible porous media, the Fourier analysis method and the priori estimates in the measure space are discussed by using the properties of the operator semigroup and the measure space and the principle of the compression image. The suitability of the solution. The existence and uniqueness of solutions for a class of non -Newtonian seepage equations with nonlinear boundary conditions are proved. Through the diffusion regularization method and the -limit compact method, we study the overall well -posedness of the solution of the equation in space when the initial value is sufficient small.