Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 3, June 2022



Page(s)  189  194  
DOI  https://doi.org/10.1051/wujns/2022273189  
Published online  24 August 2022 
Mathematics
CLC number: O193
A PDE Approach to the LongTime Asymptotic Solutions of Contact HamiltonJacobi Equations
School of Mathematics Science, Suzhou University of Science and Technology, Suzhou
215009, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: lixia0527@188.com
Received:
14
October
2021
We study the longtime asymptotic behaviour of viscosity solutions of the HamiltonJacobi equation in with a PDE approach, where is coercive in , nondecreasing in and strictly convex in , and establish the uniform convergence of to an asymptotic solution as . Moreover, is a viscosity solution of HamiltonJacobi equation .
Key words: asymptotic solution / HamiltonJacobi equation / PDE approach
Biography: WANG Yujie, female, Master candidate, research direction: mathematical methods in mechanics. Email: 757727317@qq.com
Foundation item: Supported by the National Natural Science Foundation of China (11971344) and Jiangsu Graduate Science Innovation Project (KYCX202746)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
In this paper, we consider the Cauchy problem for the evolutionary HamiltonJacobi equation
Here is an unknown function on , is a given function, and , . We study the longtime asymptotic behavior of the viscosity solution to (CP) and furthermore, discuss the relation between the limit of the viscosity solution of (CP) and the viscosity solution of the stationary HamiltonJacobi equation
There has been much study about the longtime behavior of the viscosity solutions of HamiltonJacobi equations either by means of dynamical techniques or by PDE methods. We occasionally suppress "viscosity" for simplicity.
The dynamical approach is based on the weak KAM theory initiated by Fathi^{[1,2]}. It needs strong regularity assumptions on the Hamiltonian (regularity, strict convexity and superlinearity in ) because it is based on the analysis of the associated Hamiltonian flow. Such flow is connected with the visc.solution of through the LaxOleinik formula. The dynamical approach has been later modified by Roquejoffre^{[3]}, Davini and Siconolfi in Ref. [4], and others.
The PDE approach is initiated by the work of Namah and Roquejoffre^{[5]}. It does not depend on the LaxOleinik formula, so it is possible to be applied to more general cases. Barles and Souganidis have obtained in Ref. [6] more general results in the case , for possible nonconvex Hamiltonians. We refer to Ref. [7] for a recent view on this approach.
In this paper, we will explore the visc. solutions'long time behavior of the HamiltonJacobi equation of contact type, in which the Hamiltonian explicitly depends on the unknown function . The contact Hamiltonian system is a natural extension to Hamiltonian system. Various applications of contact Hamiltonian dynamics has been found in many fields such as classic mechanics of dissipative system^{[8,9]}, mesoscopic dynamics^{[10]}, equilibrium statistical mechanics^{[11]}, and thermodynamics^{[12,13]}, etc. Su, Wang, and Yan first studied visc.solutions' longtime behavior of the contact HamiltonJacobi equation with implicit variational principle in Ref. [14], under Tonelli assumptions ( is , strict convexity and superlinear growth in for every , uniform Lipschitzity and monotonicity with respect to ). Their series of work are aimed at building the variational frame in the contact Hamiltonian system^{[1517]}. In the recent paper, the author has studied the longtime behavior of solutions of the contact HamiltonJacobi equations with the method combining the PDEviscosity solutions approach and dynamical approach under more general conditions ( is , strict convexity and coercive in for every , monotonicity with respect to )^{[18]}.
Motivated by abovementioned results, we will continue this direction of research on the longtime behavior of visc. solutions and we want to discuss if the conditions are necessary for the convergence in this paper. We mainly use and slightly modify the PDE approach which has been introduced by Barles, Ishii and Mitake (see Ref. [7]). The main difference is that we deal with the contact HamiltonianJacobi equation for the consideration of the effect of in the proof.
We assume that
, for all (CER)
The function is nondecreasing on , for all (MON)
(EP) with has a visc. solution (Z)
There exist positive constants , and a positive constant with , such that for all ,, if and , then .
()
There exist positive constants , and a positive constant with , such that for all , , if and , then .
()
Condition ()(()) means some kind of strict convexity of in . Indeed, if is strictly convex in , then
A condition similar to () has appeared in Ref. [1]. The difference is the strict convexity about in this paper and the strict convexity about both in the independent case in Ref. [7] and in the dependent case in Ref. [18]. The convexity in is the necessary condition for using this PDE approach, bcause we cannot fix to as we have done with the PDE approach in Ref. [18].
In this case, we can deal with the convergence problem of the Hamiltonian which is not strictly convex in in contrast to what happens in Ref.[14, 18]. Our main result is:
Theorem 1 Assume (C), (CER), (MON), (Z) and ()(()). Let , and let be the visc.solution of (CP). Then there exists such that
uniformly on .
Moreover, is a visc. solution of (EP), with .
The paper is organized as follows. In Section 1, we will give some classical results about visc.solution theory which are needed for the next proof. Based on the comparison theorem and the Perron method, we get the existence theorem of (CP) (Theorem 3). Assuming moreover (Z), is bounded and uniformly continuous on . In Section 2, we will give the proof of Theorem 1 with the condition ()(()) instead of the condition (CON) in the Ref.[18].
1 The Preliminary Results
As the basis of the existence theorem and the uniqueness theorem, we first introduce the comparison theorem.
Theorem 2^{[18]} Assume (C), (CER) and (MON). Let and be a visc. subsolution and a visc.supersolution of (CP), respectively, where . Then
for all .
Corollary 1^{[18]} If, in addition, are both visc. solutions of (CP), then we have
Theorem 3^{[18]} Assume (C), (CER) and (MON). Let . Then there exists a (unique) solution of (CP).
Theorem 4 ^{[18]} Assume (C), (CER), (MON) and (Z). Let be a visc. solution of (CP). Then is bounded and uniformly continuous on .
We will give some stability results concerning viscosity solutions.
Theorem 5^{[18]} Let be locally compact. is a family of viscosity subsolutions of (EP). Assume that is locally bounded in , then is also a visc. subsolution of (EP).
The theorems above are classical results in viscosity solution theory. We can find the proof in Refs. [1922].
2 The Main Result
In this section, we want to prove our main result.
Theorem 6 Assume (C), (CER), (MON), (Z) and ()(()). Let , and let be a visc.solution of (CP). Then there exists such that
uniformly on .
Moreover, is a visc. solution of (EP), with .
First, we reduce the result to the case . Indeed, we have
Lemma 1 If the result of Theorem 6 holds for any , then it holds for any .
This is an easy consequence of Theorem 2 and the reader can find a proof of the lemma above in Ref. [7].
Lemma 2 There exists a viscosity subsolution of (EP), with , such that
for all .
Proof Due to (Z), there exists a solution of (EP), with . Since the function is a solution of (CP), by Theorem 2 we obtain
for all , which can be written as
for all
with . If we set and , then we have
for all ,
and, by (MON), the function is a subsolution of (EP), with .
For , we define the function on , by
where is the function given by Lemma 2. We define the functions , with , by
Lemma 3 We have
for all .
Proof According to Lemma 2, for all ,
and
Theorem 7 The function is a subsolution of
where is the constant from (DSTC^{+}), and .
Proof Noting that and , then .
Fix any and , and assume that
If , then we have finished the proof. Therefore, we may assume that . We choose an so that
If , we get , and we are done. We may thus assume that .
Define the function by
Note that the function
on attains a strict maximum at , and that , and .
Now, if is an open ball of centered at with its closure contained in , we consider the function on given by
where is a large constant.
Let be a maximum point of . We can get
Next, set
We observe that
We have by the definition of , and by sending along an appropriate sequence, we can find points such that
where stands for the closure of , for instance, denotes the set of points for which there are sequences and such that and for all . Recall that and so that we have
from (5) and (6). Therefore, () ensures
Since , we have
Because of the assumption (MON), (8) and (9),
The second inequality holds since and because of (7). Therefore, we get
i.e.,
We set
for all .
Lemma 4 We have
for all .
Moreover, the convergence
is uniform in .
Proof If the convergence does not hold uniformly in , we can choose a sequence such that and for all and some constant . We may assume that for some . In view of the AscoliArzela theorem, we may assume by passing to a subsequence of if needed that
uniformly in ,
for some bounded function and .
By the stability of the subsolution property under uniform convergence, we see that is a subsolution of
in . Since and is bounded on , for every , the function attains a maximum at a point , then we have
Therefore, we know that
In particular, we have . Then, as usual in the viscosity solutions theory, we get
which, in the limits as , yields , a contradiction.
Proof of Theorem 6 under condition ( )
Let be the function defined by (1), with arbitrary .
Fix any . Because of (1), we may choose a constant so that for any , for all .
From the above, for any , we have
Thus, for any , we have
Now, since is bounded and Lipschitz continuous in , in view of the AscoliArzela theorem, we may choose a sequence and a bounded function so that
By (10) we get
for all . This is valid for all . Hence we obtain
Thus we find that the function is nondecreasing in for all .
From this we conclude that
for some function . Since is a viscosity solution of (CP), and is bounded on , we get from Theorem 5 that is a solution of (CP), and moreover, is a solution of (EP).
Fix any . By (14) there is a constant such that
and by (11) there is a such that
Therefore,
By the contraction property, we see that for any ,
which completes the proof.
Proof of Theorem 6 under condition ( )
We adjust to the form
where is chosen arbitrarily in and the constants and are from ().
Theorem 8 The function is a subsolution of
where is the constant from ( ), and .
We can prove
for all .
On the other hand, for any and ,
Hence, for any we have
We fix any test function and assume that attains a strict maximum at a point .
We can get the following conclusion like the proof of ():
for some .
Similarly, we can assume . According to assumption (MON), we have
Since , we get
by the condition ().
Also, we can prove
We can get is nonincreasing for all and enough large from (15). Since is bounded and Lipschitz continuous in , we can choose a sequence such that for some function ,
locally uniformly on .
is nonincreasing in for all , then there exists the function such that
uniformly on .
Furthermore
uniformly on .
Thus we complete the proof.
References
 Fathi A. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[J]. Comptes Rendus de I'Acadé mie des Sciences. Série I. Mathématique, 1997, 324(9): 10431046. [Google Scholar]
 Fathi A. Sur la convergence du semigroupe de LaxOleinik[J]. Comptes Rendus de I'Académie des Sciences. Série I. Mathématique, 1998, 327(3): 267270. [MathSciNet] [Google Scholar]
 Roquejoffre J M. Convergence to steady states or periodic solutions in a class of HamiltonJacobi equations[J]. Journal de Mathématiques Pures et Appliquées. Neuvième Série, 2001, 80(1): 85104. [CrossRef] [MathSciNet] [Google Scholar]
 Davini A, Siconolfi A. A generalized dynamical approach to the large time behavior of solutions of HamiltonJacobi equations[J]. SIAM Journal on Mathematical Analysis, 2006, 38(2): 478502. [CrossRef] [MathSciNet] [Google Scholar]
 Namah G, Roquejoffre J M. Remarks on the long time behaviour of the solutions of HamiltonJacobi equations[J]. Communications in Partial Differential Equations, 1999, 24(56): 883893. [CrossRef] [MathSciNet] [Google Scholar]
 Barles G, Souganidis P E. On the large time behavior of solutions of HamiltonJacobi equations[J]. SIAM Journal on Mathematical Analysis, 2000, 31(4): 925939. [CrossRef] [MathSciNet] [Google Scholar]
 Barles G, Ishii H, Mitake H. A new PDE approach to the large time asymptotics of solutions of HamiltonJacobi equations[J]. Bulletin of Mathematical Sciences, 2013, 3(3):363388. [CrossRef] [MathSciNet] [Google Scholar]
 Bravetti A, Cruz H, Tapias D. Contact Hamiltonian mechanics[J]. Annals of Physics, 2016, 376:1739. [Google Scholar]
 Marò S, Sorrentino A. AubryMather theory for conformally symplectic systems[J]. Communications in Mathematical Physics, 2017, 354(2): 775808. [CrossRef] [MathSciNet] [Google Scholar]
 Grmela M, Öttinger H. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism[J]. Physical Review E, 1997, 56(6): 66206632. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Bravetti A, Tapias D. Thermostat algorithm for generating target ensembles[J]. Physical Review E, 2016, 93(2): 022139. [CrossRef] [MathSciNet] [Google Scholar]
 Grmela M. Reciprocity relations in thermodynamics[J]. Physica A Statistical Mechanics & Its Applications, 2002, 309(34): 304328. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Rajeev S G. A HamiltonJacobi formalism for thermodynamics[J]. Annals of Physics, 2008, 323(9): 22652285. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Su X F, Wang L, Yan J. Weak KAM theory for HamiltonJacobi equations depending on unknown function[J]. Discrete and Continuous Dynamical Systems, 2016, 36(11): 64876522. [CrossRef] [MathSciNet] [Google Scholar]
 Wang K Z, Wang L, Yan J. Implicit variational principle for contact Hamiltonian systems[J]. Nonlinearity, 2017, 30(2): 492515. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Wang K Z, Wang L, Yan J. AubryMather theory for contact Hamiltonian systems[J]. Communications in Mathematical Physics, 2019, 366(3): 9811023. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Wang K Z, Wang L, Yan J. Variational principle for contact Hamiltonian systems and its applications[J]. Journal de Mathématiques Pures et Appliquées. Neuvième Série, 2019, 123(9): 167200. [CrossRef] [MathSciNet] [Google Scholar]
 Li X. Longtime asymptotic solutions of convex HamiltonJacobi equations depending on unknown functions[J]. Discrete and Continuous Dynamical Systems. Series A, 2017, 37(10): 51515162. [CrossRef] [MathSciNet] [Google Scholar]
 Barles G. Existence results for first order Hamilton Jacobi equations[J]. Annales de I'Institut Henri Poincare C, Analysis non linearire, 1984, 1(5): 325340. [NASA ADS] [Google Scholar]
 Barles G. Uniqueness and regularity results for firstorder HamiltonJacobi equations[J]. Indiana University Mathematics Journal, 1990, 39(2): 443466. [CrossRef] [MathSciNet] [Google Scholar]
 Ishii H. A short introduction to viscosity solutions and the large time behavior of solutions of HamiltonJacobi equations[J]. Lecture Notes in Mathematics, 2013, 2074: 111249. [CrossRef] [Google Scholar]
 Lions P L. Generalized Solutions of HamiltonJacobi Equations[M]. London: Pitman, 1982. [Google Scholar]
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.