Received 30 April 2014; revised 5 June 2014; accepted 15 June 2014

ABSTRACT

Let H > 0 be a constant, g ≥ 0 be a periodic function and. We consider a curvature flow equation V = κ + A in Ω, where for a simple curve, V denotes its normal velocity, κ denotes its curvature and A > 0 is a constant. [1] proved that this equation has a periodic traveling wave U, and that the average speed c of U is increasing in A and H, decreasing in max g' when the scale of g is sufficiently small. In this paper we study the dependence of c on A, H, max g' and on the period of g when the scale of g is large. We show that similar results as [1] hold in certain weak sense.

Keywords:

Curvature Flow Equation, Traveling Wave, Average Speed, Spatial Heterogeneity

1. Introduction

We study traveling waves for a curvature-driven motion of plane curves in a band domain Ω. The law of motion of the curve is given by

(1)

where is a simple, smooth curve, V denotes its normal velocity, denotes its curvature and A is a positive constant representing a driving force. The band domain Ω is defined as the following. Set

(2)

For some we define

where is a constant and for some (see Figure 1). Denote the left (resp. right) boundary of Ω by (resp.).

By a solution of (1) we mean a time-dependent simple, smooth curve in Ω which satisfies (1) and contacts perpendicularly. Equation (1) appears as a certain singular limit of an Allen-Cahn type nonlinear diffusion equation under the Neumann boundary conditions. The curve represents the interface between two different phases (see, e.g., [1] -[4] for details). In physics, chemistry and many other fields, an interface may propagate in a domain with obstacles, say, with obstacles lying in several lines. The motion of the interface between two adjacent lines is then like the propagation of in Ω in our problem. Hence the undulation of the boundary of Ω can be regarded as effect of obstacles and so it can be in any size. [1] studied the homogenization limit of this problem (as), we will consider the case where p is large.

To avoid sign confusion, the normal to the curve will always be chosen toward the upper region, and the sign of the normal velocity V and the curvature will be understood in accordance with this choice of the normal direction. Consequently, is negative at those points where the curve is concave while it is positive where the curve is convex (see Figure 1).

In the case where is expressed as a graph of a function at each time t. Let be the x-coordinates of the end points of lying on, , respectively. In other words, . Now (1) is equivalent to

(3)

with the boundary conditions

(4)

with. The condition in the definition prevent

from developing singularities near the boundary (cf. [1] ). Denote

(5)

and call the maximum opening angle of, or, of g. Then for.

Definition 1 A solution for some (also write as for simplicity) of (3)-(4) is called a periodic traveling wave if it satisfies for some. Its average speed is defined by

(6)

In [1] the authors proved that, under the condition, the problem (3)-(4) has a periodic traveling wave, it is unique under the normalization condition and

(7)

for all and x where U is defined. In addition, [1] studied the homogenization limit of the average speed c.

Theorem A (Theorem 2.3 in [1] ). Assume that. Let be the periodic traveling wave of (3)-(4) with average speed. Then

(8)

where is the constant determined uniquely by

(9)

and M is a positive constant independent of p. Moreover satisfies

(10)

Theorem A gives the dependence of c on A, H and near the homogenization limit (as). It is known that in the study of spatially heterogeneous problems, homogenization is a powerful method when the spatial heterogeneity is fine (for example, in our problem) (cf. [5] [6] ). On the contrary, the mathematical analysis is completely different and very difficult when the spatial heterogeneity is coarse (for example, is large in our problem). How does the traveling wave U and its average speed c in our problem depend on the parameters and p when p is large? This is an interesting problem in physics and is also a challenging one in mathematics. Some mathematicians from Japan and France have been working on it for several years, but yet very little is known so far. Our main purpose in this paper is to study this problem by analytic and numerical methods, and try to give some answers.

This paper is arranged as the following. In section 2 we list some notations and present our main theorem. In section 3 we prove the main theorem. In subsection 3.1 we prove that is increasing in A; in subsection 3.2 we prove that is increasing in some increasing sequence; in subsection 3.3 we prove that is increasing in some decreasing sequence. Finally, in section 4 we present some numerical simulation results, including the dependence of c on the period p of.

2. Notations and Main Results

We list some notations for convenience. For any, , and, denote

(11)

Clearly, N depends on and K₁, K₂ depend on. Finally, for any, , , denote

Here is an example, let, , , then we have

It is easily seen that

(12)

(13)

Therefore, if or holds, then.

The following is our main result.

Main Theorem. Assume and. Then

1) is strictly increasing in A;

2) if, then is strictly increasing in k, where for;

3) if, then is strictly increasing in, where N is given by (11), and

(14)

for.

We remark that 3) of the theorem mainly states the dependence of c on but not on g itself. In fact, for defined by (14), the conclusion of 3) holds for any provided (with restrictions ,), the exact shape of does not matter.

By the main theorem, is increasing in continuously varying A, but it is increasing in H and decreasing in only in weak sense, that is, the monotonicity holds only for certain sequences. It turns out that the monotonicity for continuously varying H and is very difficult. In fact, we believe that is not true when p is large. This is quite different from the case where.

3. Proof of the Main Theorem

In this section, for any two solutions and of (3)-(4), when we write or we mean that the inequality holds on the common domain where and are defined.

3.1. Proof of Main Theorem 1

Assume that. For, denote the (unique) periodic

traveling wave of (3)-(4) for, denote the x-span for each t by. Denote the time-period of by, that is,

Let be two times such that

for some. This is possible since. Define

for, where

Then satisfies

where

are both bounded functions. We show that

(15)

First by the maximum principle (see, for example, Theorem 2 in Chapter 3 in [7] ) we have

(16)

This implies that the graph of can not touch the graph of from below except on their end points. On the other hand, if the latter happens on the right boundary, that is, there exists such that, where

Then

(17)

and so

(18)

since, otherwise we have for x near by Taylor’s formula. But this contradicts (16).

Using (17), (18), the fact and using the equations of and we have

Since the normal velocity V in (1) is expressed by

we see that at the point, the normal velocity V₁ of U₁ and the normal velocity V₂ of U₂ satisfies

This means that, in a small time-interval around, the graph of U₁ moves along the boundary faster than the graph of U₂. This, however, contradicts the fact and the assumption that . This proves (15).

Now taking and in (15) and using the definitions of we have

By the fact in (7) we have. This implies that by the definition of c in (6). This proves 1) of the Main Theorem.

3.2. Dependence of c on H

In this subsection we study the dependence of c on H and prove Main Theorem 2). Since only H is varying, for simplicity, in this part we only indicate H but omit all the other parameters in the notations Ω, U, c, B, J, K_i, ∙∙∙.

Lemma 1 Assume that. Then for any, there holds

(19)

Proof Let such that. Denote

and

Then there exists such that, for,

Since we have

and so

(20)

For any, set. Denote the straight line passing and by and denote its slope by, then by (20) we have

where and. For any, since, the graph of must contact at some point in. If we denote the x-coordinate of the contact point by , then. Using (20) again we have

where. This proves (19). ,

Lemma 2 Assume that and. Then, where.

Proof Since we have by (12) and. The definition of and imply that and so

So by Lemma 1 we have

Since is even in x we have. Set. Then

(21)

On the other hand, replacing H by in the problem (3)-(4), we have a unique periodic traveling wave with average speed for this new problem. Using the fact and using a similar discussion as in subsection 3.1 we can compare with in the domain, and to conclude that moves faster than. So we obtain. This proves the lemma. ,

Proof of Main Theorem 2. Set as in the statement of 2), then

and

by. Define, then

Using Lemma 2 to we have. This proves 2) of the main theorem. ,

Remark If we take in the original problem, then to guarantee the existence of periodic traveling waves we should modify the boundary conditions a little. For example, one can consider a problem such that the curve always has a positive/negative slope on the right/left boundary. In this case, a similar discussion as in [1] shows that the problem has a unique periodic traveling wave U with average speed c. Moreover, or equivalently,. Using these results and using a similar discussion as above we can show that. In other words, the monotonic dependence of c on H is completely different from the case.

3.3. Dependence of c on g and α_g

In this subsection we study the dependence of c on g and. Similar as above, we only indicate g and but omit all the other parameters in the notations for simplicity.

First we note that a classical traveling wave solution of (3) (with a constant speed and a constant profile) is generally written in the form. Substituting this form into (3) yields

(22)

In addition, considering the normalization and the symmetry of Ω, we impose the following initial condition:

(23)

Denote the solution of (22)-(23) by.

Lemma 3 (Lemma 5.1 in) Assume that. Then the constant defined by

(24)

satisfies

(25)

The solution of (22)-(23) satisfies.

Lemma 4 Let, and. Assume that. Then

(26)

where is the average speed of the periodic traveling wave of (3)-(4) in band domain.

Proof From (7) we see that satisfies

for all. So U is an upper solution of

(27)

On the other hand, by Lemma 3 is a classical traveling wave of (27). So we can use comparison principle as in subsection 3.1 for and on the interval to conclude that. ,

Proof of Main Theorem 3. We write g and as and, respectively. For any, since

we have

and

By Lemma 1 and the definitions of and we have

So is a lower solution of

(28)

Replacing in Lemma 3 by, respectively, we know that

is a classical traveling wave of (28). So we can use comparison principle for and in the interval to conclude that

On the other hand, replacing by in Lemma 4 we have

Combining the above inequalities with (25) we have

This proves Main Theorem 3). ,

4. Some Numerical Simulation Results

In this section we present some numerical simulation figures. Figure 2 indicates that the average speed c is strictly increasing in the basic width H of the domain.

Figure 3 indicates that the average speed c is strictly decreasing in the maximum opening angle.

Figure 4 indicates that the average speed c is strictly increasing in the period p of g.

The results shown in Figure 2 and Figure 3 are partially proved in the main theorem. The dependence of c on p is very difficult, and we have no analytic result so far.

References