Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation

Abstract

It is shown that a strong solution of the Degasperis-Procesi equation possesses persistence property in the sense that the solution with algebraically decaying initial data and its spatial derivative must retain this property. Moreover, we give estimates of measure for the momentum support.

1. Introduction

Recently, Degasperis and Procesi [1] consider the following family of third order dispersive conservation laws: 𝑢_𝑡+𝑐₀𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥−𝛼²𝑢_𝑥𝑥𝑡=𝑐₁𝑢²+𝑐₂𝑢²_𝑥+𝑐₃𝑢𝑢_𝑥𝑥𝑥, (1) where 𝛼,  𝛾,  𝑐₀,  𝑐₁,  𝑐₂, and 𝑐₃ are real constants. Within this family, only three equations that satisfy asymptotic integrability condition up to third order are singled out, namely, the KdV equation 𝑢_𝑡+𝑢_𝑥+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0, (2) the Camassa-Holm equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (3) and a new equation (the Degasperis-Procesi equation, the DP equation, for simplicity) which can be written as (after rescaling) the dispersionless form [1] 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥. (4) It is worth noting that in [2] both the Camassa-Holm and DP equations are derived as members of a one-parameter family of asymptotic shallow water approximations to the Euler equations: this is important because it shows that (after the addition of linear dispersion terms) both the Camassa-Holm and DP equations are physically relevant; otherwise the DP equation would be of purely theoretical interest.

When 𝑐₁=−3𝑐₃/2𝛼² and 𝑐₂=𝑐₃/2 in (1), we recover the Camassa-Holm equation derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime, where 𝑢(𝑥,𝑡) represents the free surface above a flat bottom. There is also a geometric approach which is used to prove the least action principle holding for the Camassa-Holm equation, compared with [4]. It is worth pointing out that a fundamental aspect of the Camassa-Holm equation, the fact that it is a completely integrable system, was shown in [5, 6]. Some satisfactory results have been obtained for this shallow water equation recently, we refer the readers to see [7–19].

Although, the DP equation (4) has a similar form to the Camassa-Holm equation and admits exact peakon solutions analogous to the Camassa-Holm peakons [20], these two equations are pretty different. The isospectral problem for equation (4) is Ψ_𝑥−Ψ_𝑥𝑥𝑥−𝜆𝑦Ψ=0, (5) while for Camassa-Holm equation it is Ψ_𝑥𝑥−14Ψ−𝜆𝑦Ψ=0, (6) where 𝑦=𝑢−𝑢_𝑥𝑥 for both cases. This implies that the inside structures of the DP equation (4) and the Camassa-Holm equation are truly different. However, we not only have some similar results [21–23], but also have considerable differences in the scattering/inverse scattering approach, compared with the discussion in [5, 6] and in the paper [24].

Analogous to the Camassa-Holm equation, (4) can be written in Hamiltonian form and has infinitely many conservation laws. Here we list some of the simplest conserved quantities [20]: 𝐻₋₁=_ℝ𝑢³𝑑𝑥,𝐻₀=_ℝ𝑦𝑑𝑥,𝐻₁=_ℝ𝐻𝑦𝑣𝑑𝑥,₅=_ℝ𝑦^1/3𝑑𝑥,𝐻₇=_ℝ𝑦²_𝑥𝑦^−7/3+9𝑦^−1/3𝑑𝑥, (7) where 𝑣=(4−𝜕²_𝑥)⁻¹𝑢. So they are different from the invariants of the Camassa-Holm equation 𝐸(𝑢)=_ℝ𝑢²+𝑢²_𝑥𝑑𝑥,𝐹(𝑢)=_ℝ𝑢³+𝑢𝑢²_𝑥𝑑𝑥. (8)

Set 𝑄=(1−𝜕²_𝑥); then the operator 𝑄⁻¹ in ℝ can be expressed by 𝑄⁻¹1𝑓=𝐺∗𝑓=2_ℝ𝑒^{−|𝑥−𝑦|}𝑓𝑦𝑑𝑦. (9) Equation (4) can be written as 𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥3𝐺∗2𝑢²=0, (10) while the Camassa-Holm equation can be written as 𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥=0. (11) On the other hand, the DP equation can also be expressed in the following momentum form: 𝑦_𝑡+𝑦_𝑥𝑢=−3𝑦𝑢_𝑥𝑦=1−𝜕²_𝑥𝑢. (12) This formulation is important to motivate us to consider the measure of momentum support which is the second object of this paper, since we found that (12) is similar to the vorticity equation of the three-dimensional Euler equation for incompressible perfect fluids (𝑈 is the speed, and 𝜔 is its vorticity) 𝜔_𝑡+(𝑈⋅∇)𝜔=(𝜔⋅∇)𝑈,div𝑈=0,curl𝑈=𝜔. (13) The stretching term (𝜔⋅∇)𝑈 in (13) is similar to the term −3𝑦𝑢_𝑥 in (12).

One can follow the argument for the Camassa-Holm equation [8] to establish the following well posedness theorem for the Degasperis-Procesi equation.

Theorem 1 (see [23]). Given 𝑢(𝑥,𝑡=0)=𝑢₀∈𝐻^𝑠(ℝ), 𝑠>3/2, then there exist a 𝑇 and a unique solution 𝑢 to (4) (also (10)) such that 𝑢(𝑥,𝑡)∈𝐶[0,𝑇);𝐻^𝑠(ℝ)∩𝐶¹[0,𝑇);𝐻^𝑠−1(ℝ). (14)

It should be mentioned that due to the form of (10) (no derivative appears in the convolution term), Coclite and Karlsen [25] established global existence and uniqueness result for entropy weak solutions belonging to the class 𝐿¹(ℝ)∩𝐵𝑉(ℝ).

2. Unique Continuation

The purpose of this section is to show that the solution to (10) and its first-order spatial derivative retain algebraic decay at infinity as their initial values do. Precisely, we prove.

Theorem 2. Assume that for some 𝑇>0 and𝑠>3/2, 𝑢∈𝐶([0,𝑇];𝐻^𝑠(ℝ)) is a strong solution of the initial value problem associated with (10), and that 𝑢₀(𝑥)=𝑢(𝑥,0) satisfies that for some 𝜃>1𝑢₀(𝑥),𝜕_𝑥𝑢₀(𝑥)=𝑂𝑥^−𝜃as𝑥↑∞. (15) Then |𝑢(𝑥,𝑡)|,𝜕_𝑥𝑢(𝑥,𝑡)=𝑂𝑥^−𝜃as𝑥↑∞, (16) uniformly in the time interval [0,𝑇].

Notation. We will say that 𝑓(𝑥)=𝑂𝑥^−𝜃as𝑥↑∞iflim_𝑥→∞𝑓(𝑥)𝑥^−𝜃=𝐿, (17) where 𝐿 is a nonnegative constant.

Proof. We introduce the following notations: 𝐹(𝑢)=32𝑢²,𝑀=sup_{𝑡∈[0,𝑇]}‖𝑢(𝑡)‖_𝐻^𝑠. (18)(19) Multiplying (10) by 𝑢^2𝑝−1 with 𝑝∈𝑍⁺ and integrating the result in the 𝑥-variable, one gets ^∞_−∞𝑢^2𝑝−1𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥𝐺∗𝐹(𝑢)𝑑𝑥=0. (20) The first term in (20) is ^∞_−∞𝑢^2𝑝−1𝑢_𝑡𝑑𝑥=^∞_−∞12𝑝𝑑𝑢^2𝑝=1𝑑𝑡𝑑𝑥𝑑2𝑝𝑑𝑡^∞_−∞𝑢^2𝑝𝑑𝑥=‖𝑢(𝑡)‖^2𝑝−1_2𝑝𝑑‖𝑢(𝑡𝑑𝑡)‖_2𝑝, (21) and for the rest, we have ^∞_−∞𝑢^2𝑝−1𝑢𝑢_𝑥𝑑𝑥=^∞_−∞𝑢^2𝑝𝑢_𝑥≤𝑢𝑑𝑥_𝑥(𝑡)_∞‖𝑢(𝑡)‖^2𝑝_2𝑝,_∞−∞𝑢^2𝑝−1𝜕_𝑥𝐺∗𝐹(𝑢)𝑑𝑥≤‖𝑢(𝑡)‖^2𝑝−1_2𝑝𝜕_𝑥𝐺∗𝐹(𝑢)(𝑡)_2𝑝. (22) From the above inequalities, we get 𝑑‖𝑢(𝑡𝑑𝑡)‖_2𝑝≤𝑢_𝑥(𝑡)_∞‖𝑢(𝑡)‖_2𝑝+𝜕_𝑥𝐺∗𝐹(𝑢)_2𝑝, (23) and therefore, by Sobolev embedding theorem and Gronwall’s inequality, there exists a constant 𝑀 such that ‖𝑢(𝑡)‖_2𝑝≤‖𝑢(0)‖_2𝑝+^𝑡₀𝜕_𝑥𝐺∗𝐹(𝑢)_2𝑝𝑑𝜏𝑒^𝑀𝑡. (24) Since 𝑓∈𝐿¹(ℝ)∩𝐿^∞(ℝ) implies lim_𝑞→∞𝑓_𝑞=𝑓_∞, (25) taking the limits in (24) (note that 𝜕_𝑥𝐺∈𝐿¹ and 𝐹(𝑢)∈𝐿¹∩𝐿^∞) from (25) we get ‖𝑢(𝑡)‖_∞≤‖𝑢(0)‖_∞+^𝑡₀𝜕_𝑥𝐺∗𝐹(𝑢)_∞𝑑𝜏𝑒^𝑀𝑡. (26) We will now repeat the above arguments using the barrier function 𝜑_𝑁(𝑥)=𝑥1,𝑥≤1,_𝜃(),𝑁,𝑥∈1,𝑁_𝜃,𝑥≥𝑁, (27) where 𝑁∈ℤ⁺. Observe that for all 𝑁 we have 0≤𝜑^′_𝑁(𝑥)≤𝜃𝜑_𝑁(𝑥)a.e.𝑥∈ℝ. (28) Using notation in (18), from (10) we obtain 𝑢𝜑_𝑁𝑡+𝑢𝜑_𝑁𝑢_𝑥+𝜑_𝑁𝜕_𝑥𝐺∗𝐹(𝑢)=0. (29) Hence, as in the weightless case (26), we get 𝑢(𝑡)𝜑_𝑁∞≤𝑒^𝑀𝑡𝑢(0)𝜑_𝑁∞+𝑒^𝑀𝑡𝑡₀𝜑_𝑁𝜕_𝑥𝐺∗𝐹(𝑢)_∞𝑑𝜏. (30) A simple calculation shows that there exists 𝐶₀>0 depending only on 𝜃 such that, for any 𝑁∈ℤ⁺, 12𝜑_𝑁(𝑥)^∞_−∞𝑒^{−|𝑥−𝑦|}1𝜑_𝑁𝑦𝑑𝑦≤𝐶₀. (31) Thus, for any appropriate function 𝑓 one finds that 𝜑_𝑁𝜕_𝑥𝐺∗𝑓²(𝑥=1)2𝜑_𝑁(𝑥)^∞_−∞sgn𝑥−𝑦𝑒^{−|𝑥−𝑦|}𝑓²𝑦≤𝜑𝑑𝑦_𝑁(𝑥)2^∞_−∞𝑒^{−|𝑥−𝑦|}1𝜑_𝑁𝑦𝜑_𝑁𝑦𝑓𝑦𝑓𝑦≤𝜑𝑑𝑦_𝑁(𝑥)2^∞_−∞𝑒^{−|𝑥−𝑦|}𝜑_𝑁𝑦𝑑𝑦𝜑_𝑁𝑓_∞𝑓_∞≤𝐶₀𝜑_𝑁𝑓_∞𝑓_∞. (32) Combining with (30), we get 𝑢(𝑡)𝜑_𝑁∞≤𝐶₁𝑢₀𝜑_𝑁∞+^𝑡₀𝜑_𝑁𝑢_∞𝑑𝜏, (33) where 𝐶₁=𝐶₁(𝑀;𝑇,)>0. By Gronwall’s inequality, there exists a constant 𝐶 for any 𝑡∈[0,𝑇] such that 𝜑_𝑁𝑢_∞≤𝐶𝑢₀𝜑_𝑁∞≤𝐶𝑢₀⋅max1,𝑥^𝜃_∞. (34) Finally, taking the limit as 𝑁 goes to infinity in (34) we find that for any 𝑡∈[0,𝑇]𝑢(𝑥,𝑡)𝑥^𝜃≤𝐶𝑢₀⋅max1,𝑥^𝜃_∞. (35)
From (15), we get |𝑢(𝑥,𝑡)|=𝑂(𝑥^−𝜃) as 𝑥↑∞.
Next, differentiating (10) in the 𝑥-variable produces the equation 𝑢_𝑥𝑡+𝑢𝑢_𝑥𝑥+𝑢²_𝑥+𝜕²_𝑥3𝐺∗2𝑢²=0. (36) Again, multiplying (36) by 𝑢_𝑥^2𝑝−1,(𝑝∈ℤ⁺), integrating the result in the 𝑥-variable, and using integration by parts ^∞_−∞𝑢𝑢_𝑥𝑥𝑢_𝑥^2𝑝−1𝑑𝑥=^∞_−∞𝑢𝑢_𝑥^2𝑝12𝑝𝑑𝑥=−2𝑝^∞_−∞𝑢_𝑥𝑢_𝑥^2𝑝𝑑𝑥, (37) one gets the inequality 𝑑𝑢𝑑𝑡^𝑥(𝑡)_2𝑝≤2𝑢_𝑥(𝑡)_∞𝑢_𝑥(𝑡)_2𝑝+𝜕²_𝑥𝐺∗𝐹(𝑢)_2𝑝, (38) and therefore as before 𝑢_𝑥(𝑡)_2𝑝≤𝑢_𝑥(0)_2p+^𝑡₀𝜕²_𝑥𝐺∗𝐹(𝑢)_2𝑝𝑑𝜏𝑒^2𝑀𝑡. (39) Since 𝜕²_𝑥𝐺=𝐺−𝛿, we can use (25) and pass to the limit in (39) to obtain 𝑢_𝑥(𝑡)_∞≤𝑢_𝑥(0)_∞+^𝑡₀𝜕²_𝑥𝐺∗𝐹(𝑢)_∞𝑑𝜏𝑒^2𝑀𝑡; (40) from (36) we get 𝜕_𝑡𝑢_𝑥𝜑_𝑁+𝑢𝑢_𝑥𝑥𝜑_𝑁+𝑢_𝑥𝜑_𝑁𝑢_𝑥+𝜑_𝑁𝜕²_𝑥𝐺∗𝐹(𝑢)=0. (41) We need to eliminate the second derivatives in the second term in (41). Thus, combining integration by parts and (28), we find ^∞_−∞𝑢𝑢_𝑥𝑥𝜑_𝑁𝑢_𝑥𝜑_𝑁^2𝑝−1=𝑑𝑥_∞−∞𝑢𝑢_𝑥𝜑_𝑁^2𝑝−1𝜕_𝑥𝑢_𝑥𝜑_𝑁−𝑢_𝑥𝜑^′_𝑁=𝑑𝑥_∞−∞𝑢𝜕_𝑥𝑢_𝑥𝜑_𝑁^2𝑝2𝑝−𝑢_𝑥𝜑^′_𝑁𝑢_𝑥𝜑_𝑁^2𝑝−1𝑢(𝑡𝑑𝑥≤𝜅⋅‖)‖_∞+𝜕_𝑥𝑢(𝑡)_∞𝜕_𝑥𝑢𝜑_𝑁^2𝑝_2𝑝. (42) Since 𝜕²_𝑥𝐺=𝐺−𝛿, the argument in (32) also shows that 𝜑_𝑁𝜕²_𝑥𝐺∗𝑓²(𝑥)≤𝐶₀𝜑_𝑁𝑓_∞𝑓_∞. (43) Similarly, we get 𝑢_𝑥(𝑡)𝜑_𝑁∞≤𝐶₂𝑢_𝑥(0)𝜑_𝑁∞+^𝑡₀𝑢(𝜏)𝜑_𝑁∞𝑑𝜏, (44) where 𝐶₂=𝐶₂(𝑀;𝑇).
Then, taking the limit as 𝑁 goes to infinity, we find that for any 𝑡∈[0,𝑇]𝑢_𝑥(𝑡)𝑥^𝜃≤𝐶₂𝑢_𝑥(0)𝑥^𝜃_∞+^𝑡₀𝑢(𝜏)𝑥^𝜃_∞𝑑𝜏. (45) Since |𝑢(𝑥,𝑡)|=𝑂(𝑥^−𝜃) as 𝑥↑∞ and (15), we get 𝜕_𝑥𝑢(𝑥,𝑡)=𝑂𝑥^−𝜃,as𝑥↑∞. (46) This completes the proof.

3. Measure of Momentum Support

It is known that, for the Degasperis-Procesi equation, the momentum density 𝑦(𝑥,𝑡) with compactly supported initial data 𝑦₀(𝑥) will retain this property; that is, 𝑦(𝑥,𝑡) is also compactly supported [21]. However, the same argument for 𝑢(𝑥,𝑡) is false [21]. Note that a detailed description of solution 𝑢(𝑥,𝑡) outside of the support of 𝑦(𝑥,𝑡) is given in [26, 27]. Moreover, the exponential behavior of 𝑢 in 𝑥 outside this support is obvious. The comparison of the DP equation and the incompressible Euler equation above implies that the momentum 𝑦(𝑥,𝑡) in (12) plays a similar role as the vorticity does in (13). This motivates us to estimate the size of supp 𝑦(𝑡,⋅) for strong solutions. The approach is inspired by the work of Kim [28] and the recent work [29].

We first introduce the particle trajectory method. Let 𝑢∈𝐶([0,𝑇],𝐻³(ℝ))∩𝐶¹([0,𝑇],𝐻²(ℝ)) be a strong solution of (4) guaranteed by the well posedness Theorem 1. Let 𝑠∈[0,𝑇],𝑞(𝑡;𝛼,𝑠) be the solution of the following initial value problem: 𝑑𝑞(𝑡;𝛼,𝑠)()[]𝑞()𝑑𝑡=𝑢𝑠+𝑡,𝑞𝑡;𝛼,𝑠,𝑠,𝑠+𝑡∈0,𝑇,𝛼∈ℝ,0;𝛼,𝑠=𝛼,𝛼∈ℝ. (47) Then, 𝑞(𝑡;⋅,𝑠)∶ℝ→ℝ is an increasing diffeomorphism. It is shown [21, 23] that 𝑦𝑞(𝑡;𝑥,0),𝑡𝑞³_𝑥(𝑡;𝑥,0)=𝑦(𝑥,0); (48) this implies that the support of 𝑦 propagates along the flow. Set 𝐷(𝑡) to be the support of 𝑦(⋅,𝑡). Let 𝜓∈𝐿²(𝐷(𝑠)), and let 𝜓^𝑡∈𝐿²(𝐷(𝑠+𝑡)) be given by the following: 𝜓^𝑡𝑞(𝑡;𝛼,𝑠)=𝜓(𝛼). (49) Moreover, we also want to mention the standard argument on the first Dirichlet eigenvalue problem. Let Ω be an open interval in ℝ, and, 𝜆₁(Ω) be the first Dirichlet eigenvalue of the Laplacian on Ω. Then we have 𝜆₁(Ω)=inf𝜙^′2_𝐿2(Ω)∣𝜙∈𝐻¹₀(Ω)with𝜙_𝐿2(Ω)=1. (50) It is just (𝜋/|Ω|)² and the normalized eigenfunctions are the suitable translations of ±2|Ω|^1/2sin𝜋𝑥|Ω|. (51)

Theorem 3. Let 𝑦∈𝐶([0,𝑇];𝐻¹(ℝ))∩𝐶¹([0,𝑇];𝐿²(ℝ)) be a strong solution of (12). Let 𝐷(𝑡) be the support of 𝑦(⋅,𝑡) for 𝑡∈[0,𝑇] with its initial 𝐷(0) being connected.(I) Suppose there exists a positive constant 𝐾 such that 𝑢_𝑥(𝑥,𝑘)>−𝐾 for (𝑥,𝑡)∈ℝ×[0,𝑇]. Then |𝐷(0)|𝑒^{−(exp(5𝐾𝑇/2)‖𝑦0‖𝐿2(ℝ))𝑡}≤|𝐷(𝑡)|≤|𝐷(0)|𝑒^{(exp(5𝐾𝑇/2)‖𝑦0‖𝐿2(ℝ))𝑡}. (52)(II)𝑦₀ does not change sign or 𝑦₀(𝑥)≤0,𝑥∈−∞,𝑥₀,𝑦₀(𝑥)≥0,𝑥∈𝑥₀,∞, (53) and 𝑦₀∈𝐻¹(ℝ)∩𝐿¹(ℝ); then, for all 𝑡≥0|𝐷(0)|𝑒^{−‖𝑦0‖𝐿1𝑡(ℝ)}≤|𝐷(𝑡≤|𝐷(0𝑒)|)|^{‖𝑦0‖𝐿1𝑡(ℝ)}. (54)

Proof. (I) The relation of momenta 𝑦 and 𝑢 gives 𝑢()=1𝑥,𝑡2_ℝ𝑒^{−|𝑥−𝜉|}𝑦()𝑢𝜉,𝑡𝑑𝜉,_𝑥()=1𝑥,𝑡2_ℝsgn(𝜉−𝑥)𝑒^{−|𝑥−𝜉|}𝑦(𝜉,𝑡)𝑑𝜉. (55)(56) Then, we have by (12) and the lower bound of 𝑢_𝑥𝑑𝑑𝑡_ℝ𝑦²(𝑥,𝑡)𝑑𝑥=−5_ℝ𝑢_𝑥(𝑥,𝑡)𝑦²(𝑥,𝑡)𝑑𝑥≤5𝐾_ℝ𝑦²(𝑥,𝑡)𝑑𝑥. (57) Thus 𝑑𝑦(𝑑𝑡𝑥,𝑡)²_𝐿2≤5𝐾𝑦(𝑥,𝑡)²_𝐿2. (58) Therefore, (56), (58), and Gronwall inequality imply that 𝑢_𝑥(≤1𝑥,𝑡)2𝑦(𝑥,𝑡)_𝐿2≤12𝑒^5𝐾𝑇/2𝑦_0𝐿2. (59) On the other hand, due to Propositions A.2 and A.3, 𝜆₁(𝐷(𝑠)) is Lipschitz and differentiable almost everywhere. Moreover, we have −4𝑀₁𝜆₁(𝐷(𝑠≤𝑑))𝜆𝑑𝑠¹(𝐷(𝑠))≤4𝑀₁𝜆₁(𝐷(𝑠)). (60) Then, it follows that 𝑒^{−4𝑀1𝑠}𝜆₁(𝐷(0))≤𝜆₁(𝐷(𝑠))≤𝑒^4𝑀1𝑠𝜆₁(𝐷(0)) (61) with 𝜆₁(𝐷(𝑠))=𝜋²/|𝐷(𝑠)|². So (52) follows from (61) and (59).
(II) If 𝑦₀∈𝐻¹(ℝ)∩𝐿¹(ℝ) does not change sign, we conclude that solutions of (10) exist globally in time. Equality (56) and the conservation of ∫_ℝ𝑦(𝑥,𝑡)𝑑𝑥 yield 𝑢_𝑥(≤1𝑥,𝑡)2𝑦(𝑥,𝑡)_𝐿1(ℝ)=12𝑦₀(𝑥)_𝐿1(ℝ). (62) By similar arguments of (I), constant 𝑀₁ in (61) can be replaced by ‖𝑦₀(𝑥)‖_𝐿1(ℝ)/2; then (54) follows. If (53) is satisfied, we know that the solution of (10) exists globally in time [21, 30]. From (53) and (48), it is easy to get 𝑦(𝑥,𝑡)≤0,𝑥∈−∞,𝑞𝑥₀,𝑦()𝑞𝑥,𝑡𝑥,𝑡≥0,𝑥∈₀,𝑡,∞, (63) where we denote 𝑞(𝑡;𝑥,𝑠) with 𝑠=0 by 𝑞(𝑥,𝑡). By direct computation, we have _ℝ𝑦(𝑥,𝑡)𝑑𝑥=^∞_{𝑞(𝑥0,𝑡)}𝑦(𝑥,𝑡)𝑑𝑥−^{𝑞(𝑥0,𝑡)}_−∞𝑦(𝑥,𝑡)𝑑𝑥. (64) Next, we prove that ‖𝑦(𝑥,𝑡)‖_𝐿1(ℝ) is decreasing with respect to time. To this end, one gets, by differentiating (64) with respect to 𝑡 and integrating by parts, 𝑑𝑑𝑡_ℝ𝑦(𝑥,𝑡)𝑑𝑥=^∞_{𝑞(𝑥0,𝑡)}𝑦_𝑡()−𝑥,𝑡𝑑𝑥_{𝑞(𝑥0,𝑡)−∞}𝑦_𝑡()𝑞𝑥𝑥,𝑡𝑑𝑥−2𝑦𝑢₀,𝑡,𝑡=−^∞_{𝑞(𝑥0,𝑡)}𝑦_𝑥𝑢+3𝑦𝑢_𝑥+𝑑𝑥_{𝑞(𝑥0,𝑡)−∞}𝑦_𝑥𝑢+3𝑦𝑢_𝑥𝑞𝑥𝑑𝑥−2𝑦𝑢₀,𝑡,𝑡=−2^∞_{𝑞(𝑥0,𝑡)}𝑦𝑢_𝑥𝑑𝑥+2^{𝑞(𝑥0,𝑡)}_−∞𝑦𝑢_𝑥𝑑𝑥=𝑢²𝑞𝑥₀,𝑡,𝑡−𝑢²_𝑥𝑞𝑥₀=,𝑡,𝑡_{∞𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝑥^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝑥≤0. (65) This implies that 𝑢_𝑥(≤1𝑥,𝑡)2𝑦(𝑥,𝑡)_𝐿1(ℝ)≤12𝑦₀(𝑥)_𝐿1(ℝ). (66) Therefore, (54) follows by replacing 𝑀₁ with ‖𝑦₀(𝑥)‖_𝐿1(ℝ)/2 in (61).

Appendix

The following propositions with standard proofs are known in [29]; we list them here only for convenience of readers.

Proposition A.1. Let 𝑠,𝑠+𝑡∈[0,𝑇],𝛼∈𝐷(𝑠), and 𝜓∈𝐻¹₀(𝐷(𝑠));𝑢_𝑥 can be bounded by a constant 𝑀₁; then (a)𝑒^{−𝑀1|𝑡|}≤𝑞_𝛼(𝑡;𝛼,𝑠)≤𝑒^𝑀1|𝑡|, (A.1) (b)𝜓^′(𝛼)𝑒^{−𝑀1|𝑡|}≤𝜓^𝑡′𝑞()≤𝜓𝑡;𝛼,𝑠_′(𝛼)𝑒^𝑀1|𝑡|, (A.2) (c)𝜓_{𝐿2(𝐷(𝑠))}𝑒^{−𝑀1|𝑡|/2}≤𝜓^𝑡_{𝐿2(𝐷(𝑠+𝑡))}≤𝜓_{𝐿2(𝐷(𝑠))}𝑒^{𝑀1|𝑡|/2}. (A.3)

Proof. (a) Differentiating (47) with respect to 𝛼, we obtain 𝑑𝑞_𝑡𝑑𝛼=𝑢_𝑞𝑞_𝛼. (A.4) Since 𝑞(𝑡;⋅,𝑠)∶ℝ→ℝ is an increasing diffeomorphism, then 𝑞_𝛼>0. Combining the bound of 𝑢_𝑥, there holds −𝑀₁𝑞_𝛼≤𝑞_𝛼𝑡≤𝑀₁𝑞_𝛼. (A.5) This can be solved as (a).
(b) Differentiating (49) with respect to 𝛼 to get 𝜓^𝑡_𝑞𝑞_𝛼=𝜓^′(𝛼), (A.6) then (A.2) is a direct consequence of (A.1).
(c) Equation (49) and the definition of Sobolev norm give that 𝜓^𝑡2_{𝐿2(𝐷(𝑠+𝑡))}=_{𝐷(𝑠+𝑡)}𝜓^𝑡(𝑥)²𝑑𝑥=_𝐷(𝑠)𝜓²(𝛼)𝑞_𝛼𝑑𝛼, (A.7) where we have used the change of variable 𝑥=𝑞(𝑡;𝛼,𝑠). So (A.3) follows from (A.1).

Proposition A.2. Under the hypothesis of Theorem 3, for 𝑠,𝑠+𝑡∈[0,𝑇], lim_𝑡→0+𝜆sup¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≤4𝑀₁𝜆₁(𝐷(𝑠)),lim_𝑡→0−𝜆inf¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≥−4𝑀₁𝜆₁(𝐷(𝑠)). (A.8)

Proof. Let 𝑡>0,𝜙₁∈𝐻¹₀(𝐷(𝑠)) with ‖𝜙₁‖_{𝐿2(𝐷(𝑠))}=1 be a first normalized eigenfunction on 𝐷(𝑠). Then, for 𝜑∈𝐻¹₀(𝐷(𝑠+𝑡)) with ‖𝜑‖_{𝐿2(𝐷(𝑠+𝑡))}=1, we have 𝜆₁(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))=inf𝜑^′2_{𝐿2(𝐷(𝑠+𝑡))}−𝜙^′₁²_{𝐿2(𝐷(𝑠))}≤𝜙^𝑡_1𝐿⁻²_{2(𝐷(𝑠+𝑡))}𝜙^𝑡₁^′2_{𝐿2(𝐷(𝑠+𝑡))}−𝜙^′₁²_{𝐿2(𝐷(𝑠))}. (A.9) Furthermore 𝜙^𝑡_1𝐿⁻²_{2(𝐷(𝑠+𝑡))}𝜙^𝑡₁^′2_{𝐿2(𝐷(𝑠+𝑡))}=𝜙^𝑡_1𝐿⁻²_{2(𝐷(𝑠+𝑡))𝐷(𝑠)}𝜙^𝑡₁^′2𝑞_𝛼≤𝜙𝑑𝛼_{𝑡1𝐿−22(𝐷(𝑠+𝑡))}𝑒^3𝑀1𝑡𝜙^′₁²_{𝐿2(𝐷(𝑠))}≤𝑒^4𝑀1𝑡𝜙^′₁²_{𝐿2(𝐷(𝑠))}. (A.10) Combing (A.9) and (A.10) together yields lim_𝑡→0+𝜆sup¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≤lim_𝑡→0+𝑒sup^4𝑀1𝑡𝜙^′₁²_{𝐿2(𝐷(𝑠))}−𝜙^′₁²_{𝐿2(𝐷(𝑠))}𝑡=4𝑀₁𝜆₁(𝐷(𝑠)). (A.11) The second one follows by similar arguments for 𝑡<0.

Proposition A.3. Under the hypothesis of Theorem 3, for 𝑠,𝑠+𝑡∈[0,𝑇], lim_𝑡→0−𝜆sup¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≤4𝑀₁𝜆₁(𝐷(𝑠)),lim_𝑡→0+𝜆inf¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≥−4𝑀₁𝜆₁(𝐷(𝑠)). (A.12)

Proof. Let 𝜙₁∈𝐻¹₀(𝐷(𝑠)) with ‖𝜙₁‖_{𝐿2(𝐷(𝑠))}=1 be a first normalized eigenfunction on 𝐷(𝑠), and let 𝜙₂∈𝐿²(𝐷(𝑠)) be such that its 𝑡-transport is a normalized first eigenfunction on 𝐷(𝑠+𝑡). For 𝑡>0, using the left halves of (A.1) and (A.2) and then the right half of (A.3) we get 𝜙^𝑡₂^′2_{𝐿2(𝐷(𝑠+𝑡))}=_{𝐷(𝑠+𝑡)}𝜙^𝑡₂(𝑥)^′2=𝑑𝑥_𝐷(𝑠)𝜙^𝑡₂^′2𝑞_𝛼𝑑𝛼≥𝑒^{−3𝑀1𝑡}_𝐷(𝑠)𝜙^′₂(𝛼)²𝑑𝛼=𝑒^{−3𝑀1𝑡}𝜙₂²_{𝐿2(𝐷(𝑠))}𝜙₂𝜙₂²_{𝐿2(𝐷(𝑠))}^′2_{𝐿2(𝐷(𝑠))}≥𝑒^{−4𝑀1𝑡}𝜙^𝑡₂²_{𝐿2(𝐷(𝑠+𝑡))}𝜆₁(𝐷(𝑠))=𝑒^{−4𝑀1𝑡}𝜆₁(𝐷(𝑠)). (A.13) Hence lim_𝑡→0+𝜆inf¹(𝐷(𝑠+𝑡))−𝜆₁(𝐷(𝑠))𝑡≥lim_𝑡→0+𝑒inf^{−4𝑀1𝑡}−1𝑡𝜆₁(𝐷(𝑠))=−4𝑀₁𝜆₁(𝐷(𝑠)). (A.14) The other part is similar.

Acknowledgments

This work was partially supported by ZJNSF, under Grant nos. LQ12A01009 and LQ13A010008, and NSFC, under Grant nos. 11301394,11226176, and 11226172.