Abstract
We study the Navier-Stokes-Nernst-Planck-Poisson system modeling the flow of electrohydrodynamics. For small initial data, the global existence, uniqueness, and asymptotic stability as time goes to infinity of self-similar solutions to the Cauchy problem of this system posed in the whole three dimensional space are proved in the function spaces of pseudomeasure type.
1. Introduction
In this paper, we consider the Cauchy problem for the (normalized) Navier-Stokes-Nernst-Planck-Poisson system which governs the hydrodynamic transport of binary diffusion charge densities as follows (see [1]): Equations (1.1) and (1.2) are the momentum conservation and the mass conservation equations of incompressible flow. and denote, respectively, the velocity field, the pressure of the fluid, and the electrostatic potential, and the right-hand side term in (1.1) is the Lorentz force caused by the charged particles. Equations (1.3) and (1.4) model the balance between diffusion and convective transport of charge densities by the flow and the electric fields. and denote the charge densities of the negatively and positively charged species, respectively, hence the sign difference in front of the convective term in either equation. Equation (1.5) is the Poisson equation for the electrostatic potential , and the right-hand side is the net charge density. For simplicity, we have chosen the fluid density, viscosity, charge mobility and dielectric constant to be unit.
To start with, let us recall two special cases of (1.1)–(1.6). In the case that the flow is charge free, that is, , the system (1.1)–(1.6) reduces into the well-known Navier-Stokes equations: After the pioneering work [2], the Navier-Stokes equations (1.7) has drawn great attention of researchers for many years and a huge number of works can be found from the literature, for example, [3–8] and the references therein. If, on the other hand, the velocity field is identically vanishing, then (1.1)–(1.6) reduces into the following Nernst-Planck-Poisson equations which was formulated by W. Nernst and M. Planck at the end of the nineteenth century as a basic model for the diffusion of ions in an electrolytes (cf. [9]): In some literatures, it is also called the Debye-Hückel system (cf. [10]). It has drawn much attention of analysts during the past twenty years, and some works concerning existence of (large) weak solutions, (small) mild solutions, convergence rate estimates to stationary solutions of time-dependent solutions and other related topics can be found from the literature, cf., for example, [10–16] and the references therein.
In 2002, Jerome [17] proved that the system (1.1)–(1.6) has a unique local smooth solution for smooth initial data where he verified the local existence in Kato’s semigroup framework. In [18], by using the energy inequalities and the Schauder fixed point theorem, Schmuck established global existence of weak solutions to the system (1.1)–(1.6) in a bounded domain with homogeneous Neumann boundary conditions with initial data and , for . In [19], Ryham studied existence, uniqueness, and regularity of weak solutions of (1.1)–(1.6) in a bounded domain with no-flux boundary conditions for general initial data in and for small initial data in . The convergence to the stationary solution with a rate is also established in [19]. In our recent work [20], by using the - estimates of the heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we established local well-posedness of (1.1)–(1.6) in critical and subcritical Lebesgue spaces (i.e., , and , , ) and global well-posedness for small initial data in critical Lebesgue spaces (i.e., and is sufficiently small). For computational simulations of the problem (1.1)–(1.6), see [21–23].
The most important results stated in this paper are theorems on the global existence, uniqueness and asymptotic stability as time goes to infinity of self-similar solutions to the system (1.1)–(1.6) in the functional spaces of pseudomeasure-type. Let us recall that the solution of the system (1.1)–(1.6) is called a self-similar solution if it satisfies the following scaling invariant property: , , , and for all , , and , where It is clear that if is a solution of (1.1)–(1.5) with initial data (1.6), then, for each , also solves (1.1)–(1.5) with initial data Apparently such initial data do not belong to any Lebesgue and Sobolev spaces due to their strong singularity at as well as slow decay as . In [24], the authors found explicit formulas for a one-parameter family of stationary solutions of the three-dimensional Navier-Stokes equations with zero external force; these solutions are global but not smooth, more precisely, they are singular at the origin with a singularity of the kind for all time. Note that but (see (1.16) below for definition of this functional space). Similar phenomenon also appeared for the Nernst-Planck-Poisson equations; see [25]. This is the reason why we consider the system (1.1)–(1.6) in the pseudomeasure-type spaces. By a standard contraction argument, we establish global existence of solution for small initial data. It is worth pointing out that this solution is unique in a ball of the functional spaces in which the existence of solutions is going to be obtained. To overcome this restrictive condition, we establish a stability result in terms of a perturbation of initial data, which allow us to give a complete answer to the uniqueness problem of solution. Moreover, we establish the asymptotic stability of self-similar solutions as time goes to infinity. Here, we refer the reader to see [26–28] and the references cited there for more details related to the Navier-Stokes equations with measures as initial data.
The self-similar solution is related to an asymptotic behavior, for large time, of global solution to the system (1.1)–(1.6), and we could characterize the self-similar condition in the following way. Here, we disregard the functions and because when , , and are determined, and can be easily obtained from (1.2) and (1.5). A vector function has the self-similar property to the system (1.1)–(1.6) if and only if there exists a vector function such that , and for all and . In fact, when exists, these last equalities give the definition for , and it is straightforward to see that it is self-similar. Conversely, when the self-similar solution is given, we define , and for all . Then, the self-similar condition on turns out the expected equality between and by choosing .
Now, as a standard practice, we can reformulate the problem (1.1)–(1.6) into a system of integral equations. To this end, we first solve (1.5) to get as a functional of : in the distributional sense, where and denote the Fourier transform and the inverse Fourier transform, respectively. Next, it is convenient to eliminate the pressure by applying the Leray projector to both sides of (1.1), by (1.11), (1.1) and (1.2) can be transformed into the following equations: Recalling that is given formally by the formula ; that is, is the matrix pseudo-differential operator in with the symbol , where denotes the unit operator and is the Kronecker symbol. It is obvious that all these components are bounded, that is, Finally, by the well-known Duhamel principle, we see that the problem (1.1)–(1.6) can be further reduced into the following system of integral equations: where is the heat operator which can be regarded as the convolution with the heat kernel , and Later on we will work on this system of integral equations instead of (1.1)–(1.6).
Before giving the explicit meaning of solutions to the system (1.1)–(1.6), we define the functional spaces relevant to the existence of solutions of (1.1)–(1.6). Let us first define where is a given parameter. Since is not separable and the heat semigroup is not strongly continuous on this space but only weakly continuous, we will use the notation to denote by the space of functions which are weakly continuous as distributions with respect to . Next, we construct the solution of the system (1.1)–(1.6) with the velocity in the space equipped with the norm , and the components and in the space equipped with norm , where is a given parameter satisfying in the whole paper. For simplicity, we denote and for each part in the norm of , and the product of Banach spaces will be equipped with the norm .
Remark 1.1. Let ; for a positive parameter , we denote . It is easy to verify that, for each , Hence, the norms in and are invariant under the scaling (1.9).
Definition 1.2. The solution of (1.1)–(1.6) is a vector function with components satisfying, for some , and the following equalities hold for all :
Now, we state the main results of this paper as follows.
Theorem 1.3. Let , , and , . There exists a constant such that if , then the system (1.1)–(1.6) has a global solution in the space , and this is the unique solution under the condition where and the constant is defined by (2.23). Moreover, the solution depends continuously on initial data in the following sense: if , then one denotes by the unique solution of (1.1)–(1.6) with initial data , and , then, one has where .
Remark 1.4. In Theorem 1.3, we obtained only a partial answer to the uniqueness problem of solution; that is, under the restrictive condition (1.22), the solution of (1.1)–(1.6) is unique. For a complete answer to this problem, see Corollary 1.8 below.
Based on the uniqueness of solution in Theorem 1.3, by a standard way, we can deduce the existence of self-similar solution to the system (1.1)–(1.6).
Corollary 1.5. Assume that , , and satisfy the assumptions of Theorem 1.3. Assume that, moreover, Then, the solution constructed in Theorem 1.3 is a self-similar solution.
In order to give a complete answer to the uniqueness problem of solutions to the system (1.1)–(1.6), we will establish the following stability result.
Theorem 1.6. Let and belong to such that , where is a constant defined by (2.23), and let and be two solutions of (1.1)–(1.6) with initial conditions and , respectively. Then, one has
Remark 1.7. Theorem 1.6 implies that we can measure the difference of two solutions in terms of the difference of their initial data provided that the difference between these initial data is small enough.
The direct consequence of Theorem 1.6 is the following corollary.
Corollary 1.8. Assume that , in the distributional sense, and , . Then, there exists at most one solution of (1.1)–(1.6) with initial data .
Finally, we study the asymptotic stability of solutions in the sense proposed in [13] and developed in [27].
Theorem 1.9. Let be a sufficiently small number such that , where and are defined by (2.23) and (4.19), respectively. Assume that and satisfy the assumptions of Theorem 1.3, and let and be two global solutions of (1.1)–(1.6) with initial conditions and , respectively. Then, the following two conditions are equivalent:
Remark 1.10. As an interesting application of Theorem 1.9, we get the asymptotic stability of self-similar solutions to the system (1.1)–(1.6), namely, under the assumptions of Theorem 1.9, besides, assume that satisfies (1.24). Then, we know that the mild solution tends to the self-similar solution as time goes to infinity as long as satisfies the condition (1.26).
Notations
Let and be two real numbers; we denote if there is a universal constant , which does not depend on varying parameters of the problem, such that . We denote if and . In the rest part of this paper, we will use “” instead of “ ” for convenience.
Structure of the Paper
In Section 2, we prove Theorem 1.3 and Corollary 1.5. The purpose of Section 3 is to prove Theorem 1.6. In the last section, we present the proof of Theorem 1.9.
2. Global-in-Time Solutions
In this section, we give the proofs of Theorem 1.3 and Corollary 1.5. Thus, throughout this section, we assume that , , and .
2.1. The Proof of Theorem 1.3
Let be a fixed number such that , and let . Given , we define , where
We fulfil the proof of Theorem 1.3 through the following two lemmas.
Lemma 2.1. The map is well defined and maps into itself.
Proof. Note that although some parts of the proof were given in [27], we would rather give it for completeness. We first prove that is well defined and . From (2.1), we can denote , where
For , since , it is easy to see that
Thus, . To prove the weak continuity of with respect to , due to the properties of heat semigroup , it suffices to prove this for . For every , by applying the Plancherel formula, we obtain
This implies that
For , using (1.13) and the properties of the Fourier transform, we get
Here, we have used the fact that (see [29, Chapter 5, Section 1, (8)]). For the weak continuity of with respect to , we can prove it by a standard argument (cf. [30]). Hence,
For , using similar calculations,
Here, we have used the fact and the assumption to ensure that the integral is finite and independent of . It remains to show the weak continuity of , but this is a standard argument as we mentioned before. Hence,
Combining the above estimates (2.7)–(2.11), we see that
Next, we prove that is well defined and . Note that, from (2.2), we can denote , where
Since , as in the proof of , it can be easily seen that
Indeed, it suffices to estimate the second term in the norm of as follows:
For , we can do the same calculations to deal with the first term in the norm of , and one obtains that
To deal with the second term in the norm of , we need to calculate more. Note first the following two elementary inequalities:
Hence, taking the Fourier transform to , we get
This implies that . By a standard argument, we can prove that is weakly continuous with respect to . Hence,
For , we can do the same calculations as for and obtain
Concluding the above estimates (2.14)–(2.20), we have already proved that
Similarly, for , we can prove that
The proof of Lemma 2.1 is complete by (2.12), (2.21), and (2.22).
From Lemma 2.1, there exists a constant such that, for any and , one has Let be sufficiently small so that . If , then from (2.23) one has Now, let be a closed ball in with radius , that is, For any , from (2.24), we see that This implies that maps into itself.
Lemma 2.2. Let be as before . If , then is a contraction mapping.
Proof. Let , , and let , . Then by a similar argument as in the proof of Lemma 2.1, we obtain the following estimate: Since , we see that is a contraction mapping. By using Lemmas 2.1 and 2.2 and the Banach fixed point theorem, we know there exists a global solution of (1.1)–(1.6) in the space , and this is the unique solution satisfying the condition . It remains to show that the solution depends continuously on initial data. Let and be two solutions of (1.1)–(1.6) corresponding to initial conditions and , respectively, and Then, proceeding as in (2.27), we get Since , (2.29) yields that where . This proves Theorem 1.3.
2.2. The Proof of Corollary 1.5
Proof. On the one hand, from Theorem 1.3, we know, the system (1.1)–(1.6) admits a unique global solution with initial data . Moreover, . On the other hand, since , , and satisfy the condition (1.24), by the scaling invariance of (1.1)–(1.6), for each , the function (see (1.9)) is also a solution with the same initial data. Note that the norm of is invariant under the scaling (1.9), that is, Hence, by the uniqueness result of Theorem 1.3, the solution of (1.1)–(1.6) is self-similar.
3. Stability of Solutions
In this short section, we prove Theorem 1.6. Let us pick up any two solutions and associated with initial conditions and , respectively, lying in . As in the proof of Lemma 2.1, one has where . Similarly, Now, we subtract the integral equation (1.14) for from the analogous expression for , using the definition of the norm of and (3.1)-(3.2), we obtain Applying the definition of , (3.3) yields that Since we have assumed that , the continuity argument implies that , where is the smallest root corresponding to the following quadratic equation: From (3.4) we know that this root satisfies Since , this last inequality yields that The proof of Theorem 1.6 is complete.
4. Large Time Behavior of Solutions
We are now in a position to show Theorem 1.9 on the large time behavior of solutions to the system (1.1)–(1.6). Let and be two solutions of (1.1)–(1.6) constructed in Theorem 1.3 which correspond to initial conditions and , respectively. Let us recall that, by Lemma 2.1, there exists a constant , the radius of , such that Now, to simplify the notations, we introduce the following two auxiliary functions: We first assume that (1.26) holds. It follows immediately from Lemma 2.1 that By calculating the norm of , we can easily get For , let be a constant to be chosen later, we decompose the integral into and estimate each term separately: For , we change the variables and use the following identity: Thus, from (4.1), one has For , we can estimate it directly that Hence, it follows immediately from (4.7) and (4.8) that Now, we compute , by using the same argument, we obtain For , we change the variables , and from (4.1), For , we estimate it directly that From (4.11) and (4.12), we get Combining the above estimates (4.9) and (4.13), we have already proved that By using the analogous argument above, we can estimate and obtain the following estimate: The estimate of has exactly the same form as (4.15). Now, let In order to prove (1.27), it suffices to prove . Note that from (4.1) we know that is nonnegative and finite. Hence, by applying the Lebesgue dominated convergence theorem to (4.14), (4.15), and the same estimate as (4.15) for and using the assumption (1.26) we obtain where is defined by Thus, there exists a universal constant which may depend on such that Since we have assumed that , we can choose sufficiently small such that by the fact that . This implies by (4.19). We complete the proof of (1.27).
Conversely, we assume that (1.27) holds. Note that from (4.1) one has We need to prove (1.26). Repeat calculations similar to the proofs of (4.14) and (4.15), and, from the boundedness of and in (4.1), we can obtain the following estimate: It is obvious that is bounded and independent of , so (1.26) follows immediately from (1.27) and (4.21). This proves Theorem 1.9.
Acknowledgments
This work is supported by the China National Natural Science Foundation under the Grant no. 11171357. The authors would like to thank the anonymous referee for invaluable suggestions.