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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 632903, 9 pages
Large Time Behavior of the Vlasov-Poisson-Boltzmann System
1Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China
3Department of Foundation, Harbin Finance University, Harbin 150030, China
Received 18 May 2013; Accepted 1 July 2013
Academic Editor: Daniel C. Biles
Copyright © 2013 Li Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate , by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005). The improvement of the present paper is the removal of condition on parameter as in the work of Li (2008).
Large time behavior for the Boltzmann equation and related systems is an important topic for both physicists and mathematicians. We consider the Cauchy problem for Vlasov-Poisson-Boltzmann system in a torus : , which represents the distribution of particles, is a function of time , particle velocity , and position . The force in (1) is controlled by Poisson equation (2), which comes intrinsically by the nonequilibrium distribution of particles.
The quadratic term is the collision operator and is the corresponding cross-section. It is well-known by the conservation of mass that is a fixed constant which represents the background charge.
Without loss of generality, we can assume , . Define , , , which are functions of and by Physically, they represent the macroscopic quantities: density, bulk velocity, and temperature, respectively. It is well known that the conservation of mass, momentum, and energy holds: Here, the total energy consists of the kinetic energy , the internal heat energy , and the electric potential energy . By simple translation and dilation, , , can be normalized as
If the initial datum satisfies the conservation laws (7), then the stationary solution is a global Maxwellian , in the form of where the subscript represents the corresponding macroscopic quantities: density, bulk velocity, and temperature, respectively.
Traditional method for studying the asymptotic behavior is using linearization around local or global Maxwellian state. Without external force, Ukai  proved an exponential decay rate for the cutoff hard potential in a torus in 1974. In 1980, Caflisch  obtained a rate like for the cutoff soft potential with in a torus, where . Strain and Guo  extend Caflisch’s result in 2008 and get a convergence rate like for the very soft potential case (). The previous results all make use of the linearization.
However, by using some estimates on systems of second-order differential inequalities, Desvillettes and Villani  obtain an almost exponential convergence rate like . The result is weaker than using linearization, but the smallness assumption on initial data is removed and the conclusion holds for noncutoff collision kernels as well.
Our work is inspired by the work of Desvillettes and Villani . We extend their result for Boltzmann equation without external force to the Vlasov-Poisson-Boltzmann system.
Theorem 1. Let satisfy and let the collision operator satisfy for some , , where and are positive constants. Let be a smooth solution of the problem (1), (9), and (3), such that, for all , , and for all , , and , Then , such that, for all , the solution converges to in an almost exponential rate; that is, for any small positive constant , where depends on , , , , , , , , , , and .
Theorem 2. Under condition (10)–(13), the solution of problem (1)–(3) converges to in an almost exponential rate; that is, for any small positive constant , where depends on the constants in (10)–(13) and .
Now, we state some results on the existence of solutions of VPB system. The global existence of solutions is proved in  in a torus and [7–9] in the whole space with small perturbed initial data. The existence result in  also holds for a more general case, like the Vlasov-Maxwell-Boltzmann system.
First, denote some local Maxwellian states in forms of , , . Define , , , as follows: where stands for the mean temperature.
As we will show in Section 3, the gradient of temperature prevents from being close to for too long; the symmetric gradient of velocity prevents from being close to for long, that is, the local Maxwellians with constant temperature; and finally, the gradient of and prevents from being close to and for long. In order to estimate the distance between two distributions, we need to define functional and relative information (or relative entropy) between two distributions, which is the main measure of the distance between and the local Maxwellians.
Definition 3. Suppose and are two distributions on , s.t.: Define the H functional (negative of the entropy) and the Kullback relative information by
Proof. Define ; then since , we have where stands for a positive function between and . The last equality is obtained by using second-order Taylor expansion. By Hölder’s inequality, we have Since lies between and , notice that distributions , are nonnegative; thus . We have and (19) is obtained. Equation (20) follows directly from (7).
We now state the quantitative version of -theorem. See  for the proof.
Theorem 5 (Quantitative -Theorem). If is a smooth solution of the VPB equation (1), (2), then the H functional is nonincreasing as a function of , and the decreasing rate
is a positive definite functional.
Moreover, if the collision kernel satisfies (10), and complies with (12), then The only set that can make vanish is the local Maxwellian state.
We state some notations here for the fluency of description. Let and be matrices; let the operation . For a vector-valued function , the divergence is the elements of gradient matrix satisfy the symmetric part of is and the traceless part of is symbolized by :
We expect to estimate decay rate of the distance between and , and the distance is measured by Kullback relative information. By using conservation laws, a direct computation will show that the relative information between and can be decomposed into a purely hydrodynamic part and a purely kinetic part: where are nonnegative since is convex with the minimum zero at .
Moreover, denote ; we can further decompose into where It is easy to check that each of the previous terms is nonnegative by using Jensen’s inequality and convexity of functions .
It is easy to verify the following.
Lemma 6. Use the previously mentioned notations; then one has the following additivity roles:
Moreover, one has
Here nonnegative terms , , , are parts of the relative entropy, .
Now we assert the key lemma of the paper, which asserts the instability of hydrodynamic descriptions for .
Lemma 7. The following four second-order differential inequalities hold: Here , , , are small enough constants, and all constants are positive.
Roughly speaking, the previous inequalities show that cannot stay near local Maxwellian states. The gradient of prevents from staying close to for long; the symmetric gradient of prevents from staying close to for long; finally, the gradient of prevents from staying close to and . It left as the only stable state.
Lemma 8. Let be a smooth function of , . Then, for all multi-indexes , , and for all ,
Proof of Lemma 7. Most of the proof is similar to that in [4, 5]; the only difference is in estimating terms with . We will only prove (39) as an example of how to estimate terms with .
At the moment when , vanishes, so we only need to estimate : From (1) we have Here, and are matrix-valued and vector-valued functions, respectively, defined by Then, we obtain Also, we get
Then the equations of can be stated as follows:
From (46) and (47), we have
Note that , , are linearly independent in weighted space. Therefore, where It is easy to verify the convexity and nonnegativity of . Therefore,
When does not coincide with , we need to estimate two terms and of (42) separately. The detailed calculation can be found in [4, 5]. Also, we just emphasize the estimates for terms with here.
Notice that, when estimating , we need to control by . Substitute the Vlasov-Poisson-Boltzmann equation (1) into ; we get terms of .
(a) norm estimate of .
It is obvious that The first term is bounded by by interpolation lemma. As for the second term, since we have Hence, .
(b) norm estimate of .
Note that is a Gaussian distribution, so that times any polynomials of is integrable:
(c) norm estimate of .
Similarly as in the previous argument, times any polynomials of is integrable. Also, and are bounded by Schauder estimate because it is constrained by a Poisson equation.
Note that ; we have Here, is the constant appearing in the Poincaré inequality, which is only relevant to the domain . Thus,
(d) norm estimate of .
From the momentum and energy conservation of particle collisions, it is easy to verify that Thus, Then, using our continuity assumption (11) on and the interpolation Lemma 8, we can estimate norm of by . Therefore, we have
The rest of the proof is similar to that in . Now we complete the proof of the lemma.
Notice that there is the symmetric gradient of in (38); the next lemma can provide a method to control this term.
Lemma 9. One has the Korn-type inequality: and the following Poincaré-type inequalities: Here all constants are positive.
Lemma 10. One has estimates on damping of hydrodynamic oscillations with ,
See  or  for the proof of the previous two lemmas. Inequalities of Lemma 9 provide estimates of the right-hand side of second-order differential inequalities in Lemma 7. Lemma 10 provides the decay rate for hydrodynamic oscillations.
3. Proof of the Main Result
Use the previous lemmas; we are now ready to prove Theorem 2. The main idea is similar to that in ; for convenience of the reader, we restate the sketch of the proof and make it more complete by proving Lemma 11.
From -theorem (Theorem 5), the convergence rate of to is determined by entropy production functional . But there are many local Maxwellians, which make our entropy production functional vanish. Therefore it is impossible to get a uniform lower bound on the entropy production. To overcome this difficulty, it is natural to estimate the average value of entropy production. Suppose that
We wish to find an upper bound on a duration (it is possible since is monotone nonincreasing), such that where is fixed; say . Therefore, we have
Lemma 11. Choose that is small enough, like , if one can show where depends on and the various constants appearing in lemmas of Section 2. Then
Proof. Fix sufficiently small. Denote by . It is not hard to prove the continuity of . From the boundedness of initial data , we can denote , . It is sufficient to prove that, for all , or equivalently is uniformly bounded.
Define a sequence , such that
Correspondingly, we can define , . From the estimate of in (69), we have
Therefore, It is obvious that , as .
For any , we can find an interval such that . Now we are ready to estimate . From the monotonicity of , we have where the constant is independent of , since is fixed and can be chosen to be sufficiently small.
Therefore, it remains to prove condition (69). Detailed proof can be found in the last part of  for Vlasov-Poisson-Boltzmann equations; we only describe the idea of the proof for the completion of this paper. Consider on interval ; that is, has variation . In order to prove (69), it is sufficient to prove that the average value of on interval satisfies
Now we proceed the proof of Theorem 2 step by step.
: Subinterval of Where Is Large. From quantitative -Theorem 5, can be estimated directly on subinterval of where is large. The subinterval can be called , which means good interval. Other interval is called , bad interval.
Notice the entropy additivity rules in Lemma 6; we actually have
: Subinterval of Where Is Large. On interval , is small, while has lower bound . Then by entropy additivity rules, we must have that cannot be small.
Denote the subinterval of by where is large. Then from the Poincaré-type inequalities of Lemma 9, we have that is large. Therefore, the right hand side of (37) is large. By an argument for second-order differential inequalities (Lemma 12 of Desvillettes and Villani in ), we can conclude that either the average value of is large (so is ) or the length of interval is small enough to be absorbed. -theorem then asserts that average value is large.
: Subinterval of Where Is Large. On interval , , is small, while has lower bound . Then similarly, cannot be small.
Denote the subinterval of by where is large. Then from the Poincaré-type and Korn-type inequalities of Lemma 9, we have that is large. Therefore, the right-hand side of (38) is large. By an argument for second-order differential inequalities (Lemma 12 of Desvillettes and Villani in ), we can conclude thateither the average value of is large (so is ) or the length of interval is small enough to be absorbed. But the first line of (35) shows that must be large in average. -theorem then asserts that average value is large.
(4) : subinterval of where is large. On interval , , , is small, while has lower bound . Then similarly, cannot be small.
Denote the subinterval of by where is large. From the conservation of energy, we have Because of the Lipschitz continuity of , Since is sufficiently small in , therefore, (79) turns to be
Therefore, the right-hand side of (40) and (39) is large. By a similar argument as in previous subintervals, we can also show that average value is large. By a careful calculation to absorb all the bad intervals into good ones, we can prove that average value is large on interval . Thus, the whole proof is complete.
To conclude the paper, we remove the condition in Theorem 1 by making a crucial estimates on terms with . The main differences with previous works  are in proving Lemma 7. We also complete the gap in the last part of [4, 5] by proving Lemma 11.
Project 11001066 is supported by the Natural Science Foundation of China.
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