Periodic Solution Of Several Coupled Fluid Dynamics Models

This thesis is devoted to the study the periodic solutions of several coupled fluid dynamics models and hopes to get some valuable results.We give the wellposedness of periodic solutions for several coupled equations from bounded domain in two-dimensional space,bounded domain in high dimensional space and the whole space,respectively.Specifically,this thesis is divided into four chapters:In the first chapter,we introduce the background and recent research progress of coupled fluid dynamics models and then give the innovation points,difficulties and main ideas of this thesis.In the second chapter,we focus on a two-species chemotaxis-Stokes model with p-Laplacian diffusion.Here we mainly discuss the existence and regularity estimation of the periodic solution for the fluid dynamics model.The p-Laplacian diffusion term brings great difficulties when we prove the existence of periodic solution for our model.Therefore,in this thesis,we mainly use the two-level approximation method to overcome it.We first define a fourth-order regularization system to approximate the original system.Then we linearize each single equation to obtain the existence,regularity and other properties of the periodic solution by using the upper and lower solutions,induction and energy estimation methods.Finally,we can obtain the existence of the periodic solution for the fourth-order regularization system by using the Leray-Schauder fixed point theorem.At this time,we can let the coefficient δ in front of the fourth order term close to zero and then combine the Aubin-Lions lemma to obtain the first approximation.In addition,in order to eliminate the influence of the items ε|n1|sn1 which we added in the fourth order regularization system on the proof,we need to get a better estimate about n1.For this purpose,we use the standard Stokes semigroup theory and Moser iterative method to obtain ‖n1‖L∞ estimate.Then we can make ε close to zero and combine with Aubin-Lions lemma to get the second approximation.When δ and ε both tend to zero,we obtain the existence and regularity estimates of periodic solution for the original equation in the bounded domain of two-dimensional space.In the third chapter,we focus on the time-periodic solution about a three-phase model of viscoelastic fluid flow in the bounded domain of RN.In this chapter,we first introduce the definition of Lorentz space and some basic properties in the space.Then we give the Lp-Lq estimates for the second and fourth order cases.However,we do not have the corresponding extremum principle for higher order equations,that is to say,many methods available for second order equations cannot be used.Therefore in this chapter,we mainly use the idea of iterative limit to obtain the existence of solutions and other related properties.In the first step of the proof,we mainly use the recursive method to obtain the existence and regularity of the time-periodic solution about the integral system.For this integral system,we construct the corresponding iterative equation.As for the iterative equation,we can use Lp-Lq estimation and combine the idea of mathematical induction to obtain the boundedness and continuity for each iterative term.Then proceed to the next step we use the relevant properties of limit and convergence to obtain the existence and regularity of the time-periodic solution for this integral system.In the second step,we give the definition of a mild solution.Then we obtain the existence,uniqueness and regularity of a periodic mild solution for the original system by using the correlation estimate of Lp-Lq and a new recursive equation.The third step is to confirm that the time-periodic solution is also a mild solution,which we already obtained under the integral system.Then we can obtain its uniqueness and regularity directly.Finally we conclude that it is a strong solution.In the forth chapter,we discuss the time-periodic solution of a coupled CahnHilliard/Allen-Cahn system with temperature in the whole space.For the Lorentz space in the inclusion relationship between the two spaces cannot be obtained if q1=q2,p1<p2.Therefore we construct a new recursive relationship to deal with the main difficult item-vt.In this recursion relationship,we skillfully transform the problem of the derivative of time into the form of only the derivative of space.Although this will increase the order of the derivative of space,we can use the corresponding Lp-Lq estimate to ensure that our recursion is correct and feasible.Finally,we successfully obtain the existence,uniqueness and regularity of periodic mild solutions.