A Singular Limit Problem for the Rosenau–Korteweg-de Vries-Regularized Long Wave and Rosenau-regularized Long Wave Equations

Theorem 2.1

Assume that (2.2) and (2.4) hold. If

then there exist two sequences {εn}n∈ℕ and {βn}n∈ℕ with εn,βn→0, and a limit function

u ∈ L ∞ ⁢ ( ℝ + ; L 2 ⁢ ( ℝ ) ∩ L 4 ⁢ ( ℝ ) )

such that

and

Lemma 2.2

For each t>0,

In particular, we have

Proof.

Multiplying (2.3) by uε,β, we have

Since

integrating (2.11) on ℝ, we get

Estimate (2.8) follows from (2.4), (2.12) and an integration on (0,t).

We now prove (2.9). Due to (2.8) and the Hölder inequality,

u ε , β 2 ( t , x ) = 2 ∫ - ∞ x u ε , β ∂ x u ε , β d x ≤ 2 ∫ ℝ | u ε , β | | ∂ x u ε , β | d x ≤ 2 ∥ u ε , β ( t , ⋅ ) ∥ L 2 ⁢ ( ℝ ) ∥ ∂ x u ε , β ( t , ⋅ ) ∥ L 2 ⁢ ( ℝ ) ≤ C 0 β - 1 2 .

Therefore,

| u ε , β ⁢ ( t , x ) | ≤ C 0 ⁢ β - 1 4 ,

which gives (2.9).

Finally, we prove (2.10). Thanks to (2.8) and the Hölder inequality,

Hence,

∥ ∂ x ⁡ u ε , β ⁢ ( t , ⋅ ) ∥ L ∞ ⁢ ( ℝ ) ≤ C 0 ⁢ β - 3 4 ,

that is, (2.10). ∎

Lemma 2.3

Assume (2.5). Then, the following hold:

1. The family {uε,β}ε,β is bounded in L∞⁢(ℝ+;L4⁢(ℝ)).

2. The families {ε⁢∂x⁡uε,β}ε,β, {β⁢ε⁢∂x⁢x2⁡uε,β}ε,β, {ε⁢β⁢∂x⁢x2⁡uε,β}ε,β, {ε⁢β⁢∂x⁢x⁢x3⁡uε,β}ε,β, {β⁢β⁢∂x⁢x⁢x3⁡uε,β}ε,β and {β⁢∂x⁢x⁢x⁢x4⁡uε,β}ε,β are bounded in L∞⁢(ℝ+;L2⁢(ℝ)).

3. The families {β⁢ε⁢∂t⁢x2⁡uε,β}ε,β, {β⁢ε⁢∂t⁢x⁢x3⁡uε,β}ε,β, {β⁢β⁢ε⁢∂t⁢x⁢x⁢x4⁡uε,β}ε,β, {ε⁢β⁢β⁢∂x⁢x⁢x3⁡uε,β}ε,β, {ε⁢uε,β⁢∂x⁡uε,β}ε,β and {ε⁢ε⁢∂x⁢x2⁡uε,β}ε,β are bounded in L2⁢(ℝ+×ℝ).

Moreover,

Proof.

Let A,B,C be some positive constants which will be specified later. Multiplying (2.3) by

u ε , β 3 - A ⁢ β ⁢ ε ⁢ ∂ t ⁢ x ⁢ x 3 ⁡ u ε , β - B ⁢ ε 2 ⁢ ∂ x ⁢ x 2 ⁡ u ε , β + C ⁢ β 2 ⁢ ∂ x ⁢ x ⁢ x ⁢ x 4 ⁡ u ε , β ,

we have

We observe that

We have that

Since

from (2.17) and (2.18), it follows that

We observe that

We get

We have

Moreover,

From (2.16), (2.19)–(2.23), and an integration of (2.15) on ℝ, it follows that

Due to the Young inequality,

Hence, from (2.24),

From (2.5), we have

where D is a positive constant which will be specified later. From (2.10), (2.27) and the Young inequality, it follows that

Due to (2.9), (2.27) and the Young inequality,

Thanks to the Young inequality,

From (2.9), (2.27) and the Young inequality, it follows that

Again by (2.9), (2.27) and the Young inequality,

From (2.26), (2.28)–(2.32), we get

We search for A, B and C such that

5 2 - 2 ⁢ A - C 0 ⁢ D A > 0 , B - 2 ⁢ B 2 - D 2 ⁢ C 0 2 > 0   and   C 2 - 2 ⁢ A > 0 ,

that is,

We choose

The first inequality of (2.34) admits a solution if

25 - 32 ⁢ C 0 ⁢ D > 0 ,

that is,

The second inequality of (2.34) admits a solution if

1 - 8 ⁢ D 2 ⁢ C 0 2 > 0 ,

that is,

From (2.36) and (2.37), it follows that

Therefore, from (2.34), (2.35) and (2.38), we have that there exist 0<A1<A2 and 0<B1<B2, such that by choosing

(2.34) holds.

Now (2.33) and (2.34) give

for some K1,K2>0.

The assumptions in (2.4), (2.8) and an integration on (0,t) give

Hence,

for every t>0.

Arguing as in [4, Lemma 2.2], we have (2.13) and (2.14). ∎