It is convenient to make use of the Lagrange coordinates in order to establish the a priori estimates. Take the Lagrange coordinates transform
(3.1)
Since the conservation of total mass holds, the boundaries and are transformed into and , respectively, and the domain is transformed into . FBVP (1.1) and (2.1) is reformulated into
(3.2)
where the initial data satisfies
(3.3)
or
(3.4)
and the consistencies between the initial data and boundary conditions hold.
Next, we deduce the a priori estimates for the solution to FBVP (3.2).
Lemma 3.1 Let . Under the assumptions of Theorem 2.1, it holds for any strong solution to FBVP (3.2) that
(3.5)
Proof Taking the product of (3.2)2 with u, integrating on and using boundary conditions, we have
(3.6)
which leads to (3.5) after the integration with respect to . □
Lemma 3.2 Let . Under the assumptions of Theorem 2.1, it holds for any strong solution to FBVP (3.2) with the initial data satisfying (3.3) that
(3.7)
As the initial data satisfies (3.4), it holds
(3.8)
where satisfies and , satisfies and .
Proof Multiplying (3.2)1 by leads to
(3.9)
which gives
(3.10)
Summing (3.2)1 and (3.10), we deduce
(3.11)
Multiplying (3.11) by and integrating the result over , we have
(3.12)
Next, we prove (3.7) in the case that the initial data satisfies (3.3) firstly, to obtain (3.7). We assume a priori that there are constants so that
(3.13)
By means of (3.2)1 and the boundary condition, we have
(3.14)
which yields
(3.15)
From (3.13), we can obtain
(3.16)
It holds from (3.5), (3.13), (3.15) and (3.16) that
(3.17)
where is a positive constant independent of time, and we assume that
(3.18)
Using the same method we can obtain
(3.19)
From (3.12), (3.17) and (3.19), we have (3.7).
Then, we prove (3.8) as the initial data satisfies (3.4), where we have the fact
(3.20)
Applying equation (3.2)1 and the boundary condition, we have that
(3.21)
and
(3.22)
which together with
(3.23)
(3.24)
and
(3.25)
gives rise to (3.8). □
Lemma 3.3 Let . Under the assumptions of Theorem 2.1, it holds
(3.26)
where and are positive constants independent of time.
Proof Define
(3.27)
and
(3.28)
It is easy to verify that and . In addition, it holds as that
(3.29)
and as that
(3.30)
As the initial data satisfies (3.3), it follows from (3.5) and (3.7) that
(3.31)
where and ν are positive constants independent of time, and we assume that
(3.32)
As and as , from the condition
(3.33)
we can find that there are two positive constants and independent of time and choose
(3.34)
such that
(3.35)
As the initial data satisfies (3.4), it follows from (3.5) and (3.8) that
(3.36)
where ν is a positive constant independent of time, and we use the fact that
(3.37)
which together with Young’s inequality gives
(3.38)
where C is a positive constant independent of time. As and as , by means of the condition
(3.39)
we can obtain two positive constants and independent of time such that
(3.40)
The proof of this lemma is completed. □
We also have the regularity estimates for the solution to FBVP (3.2) as follows.
Lemma 3.4 Let . Under the assumptions of Theorem 2.1, it holds for any strong solution to FBVP (3.2) that
(3.41)
If it is also satisfied that
(3.42)
then the strong solution
has the regularities
(3.43)
Proof Multiplying (3.2)2 by , integrating the result over and making use of the boundary conditions, after a direct computation and recombination, we have
(3.44)
Integrating equation (3.44) over , from (3.5), (3.7), (3.8) and (3.26), it is easily verified that
(3.45)
where C denotes a constant independent of time. From (3.2)2, (3.5), (3.7), (3.8) and (3.26), we deduce
(3.46)
Using (3.46), we can obtain that
(3.47)
which together with (3.2)2 implies
(3.48)
Differentiating (3.2)2 with respect to τ, we get
(3.49)
Taking product between (3.49) and , integrating the results over and using the boundary conditions (3.2)4, we have
(3.50)
The terms on the right-hand side of (3.50) can be bounded respectively as follows:
(3.51)
and
(3.52)
Summing (3.50)-(3.52) together and making use of (3.26) and (3.48), we obtain
(3.53)
Integrating equation (3.53) over , we get from (3.2)2, (3.5) and (3.26) that
(3.54)
which gives
(3.55)
which implies , and it follows from the definition of and that . The proof of this lemma is completed. □
Finally, we give the large time behaviors of the strong solution as follows.
Lemma 3.5 Let . Under the assumptions of Theorem 2.1, it holds for any strong solution to FBVP (3.2) that
(3.56)
where and denote two positive constants independent of time.
Proof Applying (3.6) and (3.12) with modification, we can obtain
(3.57)
and
(3.58)
We have from (3.5), (3.7), (3.26), the Gagliardo-Nirenberg-Sobolev inequality and Young’s inequality that
(3.59)
(3.60)
and define
(3.61)
As the initial data satisfies (3.3), we have
(3.62)
where C and are positive constants independent of time. By (3.57)-(3.62), a complicated computation gives rise to
(3.63)
where is a positive constant independent of time. From (3.63), we have
(3.64)
As the initial data satisfies (3.4), from (3.22)-(3.24), we obtain
(3.65)
which implies that the domain expands as t grows up, so that we have
(3.66)
Then it holds from (3.65) and (3.66) that
(3.67)
Using (3.58), after a complicated computation, we have
(3.68)
where is a positive constant independent of time, and we have
(3.69)
where C is a positive constant independent of time.
By the fact
(3.70)
where is a constant independent of time, and the Gagliardo-Nirenberg-Sobolev inequality
(3.71)
we can deduce (3.56). □