Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
-
Yukun Song
and
Fengming Liu
Abstract
This paper is concerned with a compressible shear thickening fluid-particle interaction model for the evolution of particles dispersed in a viscous non-Newtonian fluid. Taking the influence of non-Newtonian gravitational potential into consideration, the existence and uniqueness of strong solutions are established.
1 Introduction
We consider a compressible non-Newtonian fluid-particle interaction model which reads as follows
with the initial and boundary conditions
and the no-flux condition for the density of particles
where ρ, u, η, P(ρ) = aργ denote the fluid 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 > 0, γ > 1, μ1 > 0, p > 2, 1 < q < 2, λ > 0 is the viscosity coefficient and β ≠ 0 is a constant. Ω is a one-dimensional bounded interval, for simplicity we only consider Ω = (0, 1), ΩT = Ω × [0, T].
In fact, there are extensive studies concerning the theory of strong and weak solutions for the multi-dimensional fluid-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 fluid-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 fluid-particle interaction model in 2D case followed the spirit of reference [6].
The non-Newtonian fluid is an important type of fluid because of its immense applications in many fields of engineering fluid mechanics such as inks, paints, jet fuels etc., and biological fluids such as blood (see [7]). Many researchers turned to the study of this type of fluid under different 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 fluid-interaction model systems with non-Newtonian gravitational potential. There are still no existence results to problem (1)-(3) when p > 2, 1 < q < 2 which describes that the motion of the compressible viscous isentropic gas flow 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 difficulties 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 = 1. In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as
We state the definition of strong solution as follows:
Definition 1.1
The (ρ, u, Φ, η) is called a strong solution to the initial boundary value problem (1)-(3), if the following conditions are satisfied:
For all φ ∈ L∞(0, T*;H1(Ω)), φt ∈ L∞(0, T*;L2(Ω)), for a.e. t ∈ (0, T), we have
For all
For allϑ ∈ L∞(0, T*;H2(Ω)), ϑt ∈ L∞(0, T*H1(Ω)), for a.e. t∈ (0, T), we have
For all ψ ∈ L∞(0, T*;H2(Ω)), ψt ∈ L∞(0, T*;L2(Ω)), for a.e. t ∈ (0, T), we have
1.1 Main results
Theorem 1.2
Let μ1 > 0 be a positive constant and Φ ∈ C2(Ω), and assume that the initial data (ρ0, u0, η0) satisfy the following conditions
and the compatibility condition
for some g ∈ L2(Ω). Then there exist a T*∈(0, +∞) and a unique strong solution (ρ, u, η) to(1)-(3)such that
2 A priori Estimates for Smooth Solutions
In this section, we will prove the local existence of strong solutions. By virtue of the continuity equation (1)1, we deduce the conservation of mass
Provided that (ρ, u, η) is a smooth solution of (1)-(3) and ρ0 ≥ δ, where 0 < δ ≪ 1 is a positive number. We denote by
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.
2.1 Estimate for
| u | W 0 1 , p
By using (1)1, we rewrite the (1)2 as
Multiplying (10) by ut, integrating (by parts) over ΩT, we have
We deal with each term as follows:
Since from (1)1 we get
Substituting the above into (11), we obtain
Using Young’s inequality, we obtain
On the other hand, multiplying (1)3 by Ψ and integrating over Ω, we get
Then we have
Differentiating (1)3 with respect to x, multiplying it by Ψx and integrating over Ω, we have
By virtue of
and
Therefore,
We deal with the term of |uxx|L2. Notice that
Then
Taking the above inequality by L2 norm, we get
Hence, we deduce that
Moreover, using (1)1, we have
where C is a positive constant, depending only on M0.
2.2 Estimate for |ρ|H1
From (1)3, taking it by L2 norm, we get
Multiplying (1)1 by ρ, integrating over Ω, we have
Integrating by parts, using Sobolev inequality, we deduce that
Differentiating (1)1 with respect to x, and multiplying it by ρx, integrating over Ω, using Sobolev inequality, we have
From (19) and (20), by Gronwall’s inequality, it follows that
Besides, by using (1)1, we can also get the following estimates:
2.3 Estimate for |ηt|L2 and |η|H1
Multiplying (1)4 by η, integrating the resulting equation over ΩT, using the boundary conditions (3), Young’s inequality, we have
Multiplying (1)4 by ηt, integrating (by parts) over ΩT, using the boundary conditions (3), Young’s inequality, we have
Differentiating (1)4 with respect to t, multiplying the resulting equation by ηt, integrating (by parts) over ΩT, we get
2.4 Estimate for
| ρ u t | L 2
Differentiating equation (10) with respect to t, multiplying the result equation by ut, and integrating it over Ω with respect to x, we have
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 Ij as follows
where C is a positive constant, depending only on M0.
Next, we deal with the term |Ψxt|L2 of I12. Differentiating (1)3 with respect to t, multiplying it by Ψt, integrating over Ω and using Young’s inequality, we obtain
By virtue of
and
then
Substituting Ij(j = 1, 2, …, 12) into (28), and integrating over (τ, t) ⊂ (0, T) on the time variable, we have
To obtain the estimate of
According to the smoothness of (ρ, u, η), we obtain
Therefore, taking a limit on τ in (29), as τ → 0, we conclude that
where C is a positive constant, depending only on M0.
Combining the estimates of (15), (18), (21), (22), (17), (26), (30) and the definition of Z(t), we conclude that
where c͂, c͂͂ are positive constants, depending only on M0. This means that there exist a time T1 > 0 and a constant C > 0, such that
3 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: first define u0 = 0 and assuming that uk−1 was defined for k ≥ 1, let ρk, uk, ηk 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
where C is a generic constant depending only on M0, but independent of k.
In addition, we first find ρk from the initial problem
with smooth function uk−1, obviously, there is a unique solution ρk to the above problem and also by a standard argument, we could obtain that
Next, we have to prove that the approximate solution (ρk, uk, ηk) converges to a solution to the original problem (1) in a strong sense. To this end, let us define
then we can verify that the functions ρk+1, uk+1, ηk+1 satisfy the system of equations
Multiplying (34) by ρk+1, integrating over Ω and using Young’s inequality, we obtain
where Cζ is a positive constant, depending on M0 and ζ for all t < T1 and k ≥ 1.
Multiplying (35) by uk+1, integrating over Ω and using Young’s inequality, we obtain
Let
then
We estimate the second term of (39) as follows
Similarly, multiplying (36) by Ψk+1, integrating over Ω, we get
since
and
Then
That means (41) turns into
Substituting (40) and (42) into (39), using Young’s inequality, yields
where
Multiplying (37) by ηk+1, integrating over Ω, using (33) and Young’s inequality, we have
Collecting (38), (43) and (44), we obtain
where Eζ(t) depends only on Bζ(t) and Cζ, for all t ≤ T1 and k ≥ 1. Using (33), we have
Integrating (45) over (0, t) ⊂ (0, T1) with respect to t, using Gronwall’s inequality, we have
From the above recursive relation, choose ζ > 0 and 0 < T* < T1 such that C exp
Since all of the constants do not depend on δ, as k → ∞, we conclude that sequence (ρk, uk, ηk) converges to a limit (ρδ, uδ, ηδ) in the following convergence
and there also holds
For each small δ > 0, let
where
We deduce that (ρδ, uδ, ηδ) is a solution of the following initial boundary value problem
where
By the proof of Lemma 2.3 in [11], there exists a subsequence
where C is a positive constant, depending only on M0.
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.
Acknowledgement
The authors would like to thank the anonymous referees for their valuable suggestions and comments which improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (Nos.11526105; 11572146), the founds of education department of Liaoning Province (Nos.JQL201715411; JQL201715409).
References
[1] Carrillo J. A., Karper T., Trivisa K., On the dynamics of a fluid-particle model: the bubbling regime, Nonlinear Analysis, 2011, 74, 2778-2801.10.1016/j.na.2010.12.031Search in Google Scholar
[2] Fang D. Y., Zi R. Z., Zhang T., Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys. 2012, 53, 033706.10.1063/1.3693979Search in Google Scholar
[3] Ballew J., Trivisa K., Suitable weak solutions and low stratification singular limit for a fluid particle interaction model. Quart. Appl. Math., 2012, 70, 469-494.10.1090/S0033-569X-2012-01310-2Search in Google Scholar
[4] Ballew J., Trivisa K., Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system. Nonlinear Analysis, 2013, 91, 1-19.10.1016/j.na.2013.06.002Search in Google Scholar
[5] Constantin P., Masmoudi N., Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, Commun. Math. Phys., 2008, 278, 179-191.10.1007/s00220-007-0384-2Search in Google Scholar
[6] Chemin J. Y. and Masmoudi N., About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 2001, 33, 84-112.10.1137/S0036141099359317Search in Google Scholar
[7] Chhabra R. P., Bubbles, Drops, and Particles in Non-Newtonian Fluids, 2nd ed. (Taylor & Francis, New York, 2007).10.1201/9781420015386Search in Google Scholar
[8] Ladyzhenskaya O. A., New equations for the description of viscous incompressible fluids and solvability in the large of the boundary value problems for them. In Boundary Value Problems of Mathematical Physics, vol. V, Amer. Math. Soc., Providence, RI (1970).Search in Google Scholar
[9] Málek J., Nečas J., Rokyta M., Ru̇žička M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall, New York (1996).10.1007/978-1-4899-6824-1Search in Google Scholar
[10] Mamontov A. E., Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Ann. Univ. Ferrara-Sez. VII-Sc. Mat, 2000, 139-160.10.1007/BF02674554Search in Google Scholar
[11] Yuan H. J. and Xu X. J., Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differential Equations, 2008, 245, 2871-2916.10.1016/j.jde.2008.04.013Search in Google Scholar
[12] Rozanova O., Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space, J. Math. Anal. Appl., 2010, 371, 190-194.10.1016/j.jmaa.2010.05.013Search in Google Scholar
[13] Cho Y., Choe H., Kim H., Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl. 2004, 83, 243-275.10.1016/j.matpur.2003.11.004Search in Google Scholar
[14] Cho Y., Kim H., Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 2006, 228, 377-411.10.1016/j.jde.2006.05.001Search in Google Scholar
© 2018 Song and Liu, published by De Gruyter
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
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- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
