Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential

https://doi.org/10.1515/math-2018-0064 Received February 15, 2018; accepted May 6, 2018. Abstract: This paper is concerned with a compressible shear thickening uid-particle interaction model for the evolution of particles dispersed in a viscous non-Newtonian uid. Taking the in uence of non-Newtonian gravitational potential into consideration, the existence and uniqueness of strong solutions are established.


Introduction
We consider a compressible non-Newtonian uid-particle interaction model which reads as follows with the initial and boundary conditions (ρ, u, η) t= = (ρ , u , η ), and the no-ux condition for the density of particles where ρ, u, η,P(ρ) = aρ γ denote the uid density, velocity, the density of particle in the mixture and pressure respectively, Ψ denotes the non-Newtonian gravitational potential and the given function Φ(x) denotes the external potential. a > , γ > , µ > , p > , < q < , λ > is the viscosity coe cient and β ≠ is a constant. Ω is a one-dimensional bounded interval, for simplicity we only consider Ω = ( , ) , In fact, there are extensive studies concerning the theory of strong and weak solutions for the multidimensional uid-particle interaction models for the newtonian case. In [1], Carrillo et al. discussed the global existence and asymptotic behavior of the weak solutions for a uid-particle interaction model. Subsequently, Fang et al. [2] obtained the global classical solution in dimension one. In dimension three, Ballew and Trivisa [3,4] established the global existence of weak solutions and the existence of weakly dissipative solutions under reasonable physical assumptions on the initial data. In addition, Constantin and Masmoudi [5] obtained the global existence of weak solutions for a coupled incompressible uid-particle interaction model in 2D case followed the spirit of reference [6].
The non-Newtonian uid is an important type of uid because of its immense applications in many elds of engineering uid mechanics such as inks, paints, jet fuels etc., and biological uids such as blood (see [7]). Many researchers turned to the study of this type of uid under di erent conditions both theoretically and experimentally. For details, we refer the readers to [8][9][10][11][12] and the references therein. To our knowledge, there seems to be a very few mathematical results for the case of the uid-interaction model systems with non-Newtonian gravitational potential. There are still no existence results to problem (1)-(3) when p > , < q < which describes that the motion of the compressible viscous isentropic gas ow is driven by a non-Newtonian gravitational force.
We are interested in the existence and uniqueness of strong solutions on a one dimensional bounded domain. The strong nonlinearity of (1) bring us new di culties in getting the upper bound of ρ and the method used in [2] is not suitable for us. Motivated by the work of Cho et al. [13,14] on Navier-Stokes equations, we establish local existence and uniqueness of strong solutions by the iteration techniques.
Throughout the paper we assume that a = λ = . In the following sections, we will use simpli ed notations for standard Sobolev spaces and Bochner spaces, such as We state the de nition of strong solution as follows: is called a strong solution to the initial boundary value problem (1)- (3), if the following conditions are satis ed: (v) For all ψ ∈ L ∞ ( , T * ; H (Ω)), ψ t ∈ L ∞ ( , T * ; L (Ω)), for a.e. t ∈ ( , T), we have . Main results Theorem 1.2. Let µ > be a positive constant and Φ ∈ C (Ω), and assume that the initial data (ρ , u , η ) satisfy the following conditions and the compatibility condition for some g ∈ L (Ω). Then there exist a T * ∈ ( , +∞) and a unique strong solution (ρ, u, η) to (1)- (3) such that

A priori Estimates for Smooth Solutions
In this section, we will prove the local existence of strong solutions. By virtue of the continuity equation ( ) , we deduce the conservation of mass Provided that (ρ, u, η) is a smooth solution of (1)-(3) and ρ ≥ δ, where < δ ≪ is a positive number. We denote by M = + µ + µ − + ρ H + g L , and introduce an auxiliary function Then we estimate each term of Z(t) in terms of some integrals of Z(t), apply arguments of Gronwall-type and thus prove that Z(t) is locally bounded.

. Estimate for u W ,p
By using ( ) , we rewrite the ( ) as Multiplying (10) by u t , integrating (by parts) over Ω T , we have We deal with each term as follows: Since from ( ) we get Substituting the above into (11), we obtain Using Young's inequality, we obtain On the other hand, multiplying ( ) by Ψ and integrating over Ω, we get Di erentiating ( ) with respect to x, multiplying it by Ψ x and integrating over Ω, we have By virtue of Therefore, We deal with the term of u xx L . Notice that Then Taking the above inequality by L norm, we get Hence, we deduce that Moreover, using ( ) , we have Combining (13) where C is a positive constant, depending only on M .
. Estimate for ρ H From ( ) , taking it by L norm, we get Multiplying ( ) by ρ, integrating over Ω, we have Integrating by parts, using Sobolev inequality, we deduce that Di erentiating ( ) with respect to x, and multiplying it by ρ x , integrating over Ω, using Sobolev inequality, we have d dt From (19)and (20), by Gronwall's inequality, it follows that Besides, by using ( ) , we can also get the following estimates: . Estimate for η t L and η H Multiplying ( ) by η, integrating the resulting equation over Ω T , using the boundary conditions (3), Young's inequality, we have Multiplying ( ) by η t , integrating (by parts) over Ω T , using the boundary conditions (3), Young's inequality, we have Di erentiating ( ) with respect to t, multiplying the resulting equation by η t , integrating (by parts) over Ω T , we get Combining (23)-(25), we get . Estimate for √ ρu t L Di erentiating equation ( ) with respect to t, multiplying the result equation by u t , and integrating it over Ω with respect to x, we have d dt Note that

Combining (12), (27) can be rewritten into
By using Sobolev inequality, Hölder inequality and Young's inequality, (14),(15), we estimate each term of I j as follows where C is a positive constant, depending only on M . Next, we deal with the term Ψ xt L of I . Di erentiating ( ) with respect to t, multiplying it by Ψ t , integrating over Ω and using Young's inequality, we obtain By virtue of (t). Therefore, Substituting I j (j = , , . . . , ) into (28), and integrating over (τ , t) ⊂ ( , T) on the time variable, we have To obtain the estimate of √ ρu t (t) L , we need to estimate lim τ → √ ρu t (τ ) L . Multiplying (10) by u t and integrating over Ω, we have According to the smoothness of (ρ, u, η), we obtain Therefore, taking a limit on τ in (29), as τ → , we conclude that where C is a positive constant, depending only on M .
Combining the estimates of ( ), ( ), ( ), ( ), ( ), ( ), ( ) and the de nition of Z(t), we conclude that whereC,C are positive constants, depending only on M . This means that there exist a time T > and a constant C > , such that ess sup

Proof of the Main Theorem
In this section, our proof will be based on the usual iteration argument and some ideas developed in [13,14]. Precisely, we construct the approximate solutions, by using the iterative scheme, inductively, as follows: rst de ne u = and assuming that u k− was de ned for k ≥ , let ρ k , u k , η k be the unique smooth solution to the following problems: with the initial and boundary conditions with the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold ess sup where C is a generic constant depending only on M , but independent of k.
In addition, we rst nd ρ k from the initial problem with smooth function u k− , obviously, there is a unique solution ρ k to the above problem and also by a standard argument, we could obtain that Next, we have to prove that the approximate solution (ρ k , u k , η k ) converges to a solution to the original problem ( ) in a strong sense. To this end, let us de nē then we can verify that the functionsρ k+ ,ū k+ ,η k+ satisfy the system of equations Multiplying (34) byρ k+ , integrating over Ω and using Young's inequality, we obtain where C ζ is a positive constant, depending on M and ζ for all t < T and k ≥ . Multiplying (35) byū k+ , integrating over Ω and using Young's inequality, we obtain d dt Let We estimate the second term of (39) as follows Similarly, multiplying (36) byΨ k+ , integrating over Ω, we get since Then That means (41) turns into Substituting (40) and (42) into (39), using Young's inequality, yields where B ζ (t) = C( + u k xt (t) L ), for all t ≤ T and k ≥ . Using (33) we derive t B ζ (s)ds ≤ C + Ct.
We deduce that (ρ δ , u δ , η δ ) is a solution of the following initial boundary value problem where ρ δ ≥ δ, p > , < q < . By the proof of Lemma 2.3 in [11], there exists a subsequence {u where C is a positive constant, depending only on M .
The uniqueness of solution can be obtained by the same method as the above proof of convergence, we omit the details here. This completes the proof.