Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 6905269 | https://doi.org/10.1155/2020/6905269

Zenggui Wang, "Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 6905269, 12 pages, 2020. https://doi.org/10.1155/2020/6905269

Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

Academic Editor: Youssef N. Raffoul
Received11 Nov 2019
Accepted18 Feb 2020
Published19 Mar 2020

Abstract

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.

1. Introduction

In this paper, we study hyperbolic inverse mean curvature flow (HIMCF):where k denotes the mean curvature of the curve γ and and are, respectively, the unit inner normal and tangent vectors of the curve γ (·, t). γ0 stands for a smooth strictly convex closed curve, γ1 denotes the initial velocity of γ0, and is the unit inner normal vector of γ0.

Definition 1. (see [1]). A flow is called a normal preserving flow if the normal vector of the curve is independent of time , i.e., holds for all time.
It is well known that inverse mean curvature flow is an important method to derive the energy estimates in general relativity; for example, Huisken and Ilmanen developed a theory of weak solutions of the inverse mean curvature flow and used it to prove successfully Riemannian Penrose inequality which plays an important role in general relativity (see [2]). In [3], Urbas proved that, for inverse mean curvature flow, the surfaces stay strictly convex and smooth for all time. Furthermore, the surfaces become more and more spherical in the process. Similar results have also been obtained for star-shaped initial data with the positive mean curvature of the surface, see Urbas [3] and Gerhardt [4]. In [2], Huisken and Ilmanen proved a sharp lower bound of mean curvature, from which they proved that if the initial surface is the boundary of a strictly star-shaped domain and has nonnegative mean curvature, a smooth solution of the inverse mean flow will exist for all time and converge to a manifold.
The hyperbolic mean curvature flow, i.e., the hyperbolic version of mean curvature flow, has been introduced by some authors, see Gurtin and Podio-Guidugli [5], He et al. [6], Kong et al. [7, 8], Lefloch and Smoczyk [9], Notz [10], Rotstein et al. [11], and Wang [12, 13]. In fact, Yau in [14] has suggested the following equation related to a vibrating membrane or the motion of a surface:where H is the mean curvature and is the unit inner normal vector of the surface and pointed out that very little is known about the global time behavior of the hypersurfaces. Recently, in [1], Chou and Wo proposed a new hyperbolic curvature flow for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. The equation satisfied by the graph of the hypersurface under this flow gives rise to a new class of fully nonlinear Euclidean invariant hyperbolic equations. Finally, they consider the expanding Gauss curvature flow driving by negative powers −kβ of the Gauss curvature. In the special case β = 1, the expanding flow becomes, once written in terms of the support function S(θ, t), a linear problem:where S(θ, t) = {γ(θ, t), (cos θ, sin θ)}, and the normal angle θ is determined modulo 2π. They get the following result.

Proposition 1. (see [1]). Consider (3) where the initial values are smooth and satisfy ϕθθ + ϕ > 0 and ψθθ + ψ > 0. Then, the flow remains smooth and expands to infinity like a circle.
Motivated by the inverse mean curvature flow and the local solution theory of the Cauchy problem for the quasilinear hyperbolic system (see [15]), we will focus on life span (the maximum existence time of unique local classical solutions) of classical solutions to the Cauchy problem for the hyperbolic inverse mean curvature flow. Our first result is the following local existence theorem for initial value problem (1).

Theorem 1. (local existences and uniqueness). Suppose that γ0 is a smooth, strictly convex curve, and is a smooth vector function. Then, there exist positive T and a family of strictly convex closed curves γ(·, t) with t ∈ [0, T) such that γ(·, t) satisfies (1).
The following theorem concerns the life-span of local (in space) smooth solutions of flow (1) that can be written as convex graphs over an interval in the formWriting the initial conditions asfor some and , we obtain the following result.

Theorem 2. Assume that γ0 is a strictly convex closed curve and is a smooth vector function on S1. Then, a lower bound δ for the maximum time T of existence of a solution of (1) in the form G is given bywhere , ,The paper is organized as follows. In Section 2, we derive a hyperbolic Monge–Ampère equation by the hyperbolic inverse mean curvature flow and give the short-time existence theorem, i.e., Theorem 1. In Section 3, we reduce the hyperbolic Monge–Ampère equation to a quasilinear hyperbolic system in Riemann invariants. Section 4 is devoted to prove the main result, i.e., Theorem 2, by the local solution of theory for the Cauchy problem of the quasilinear hyperbolic system.

2. Hyperbolic Monge–Ampère Equation

Suppose for z ∈ (a, b) and t ∈ (t0, t1), the curve γ(z, t) can be expressed in the form of a graph (x, u(x, t)), and we have

Taking the inner product with the choice and , respectively, yields

It would be convenient to write the expression of the curvature k for graphs:

Flow (1) is a normal preserving flow. First, we need to compute . Using the identitiesone computes

This is an ODE of the form . Clearly, (1) is a normal preserving flow. Then, having observed that also the tangent vector is a constant, by using it, we can easily to show the identity

By , the normal preserving flow is reducible, and the curvature k for graphs , graph form of flow (1), satisfies the following equation:

This is a fully nonlinear hyperbolic Monge–Ampère equation as long as the curve is uniformly convex, i.e., flow (1) reduces to

Remark 1. We have expressed flow (1) as the equation of support function in (3). When the curve γ is represented as a graph and described by the support function simultaneously, the following relations hold:For an unknown function h = h(x, t) defined on , Monge–Ampère equation iswhere the coefficients A, B, C, D, and E depend on t, x, h, ht, hx. We say that (18) is a hyperbolic Monge–Ampère equation ifInitial conditions for the Cauchy problem on the Ox axis arewhere and . Suppose h0 and h1 satisfy the next two conditions. First, the axis Ox is free, i.e.,Secondly, on the axis Ox, the hyperbolic conditionholds.
It is easy to prove that (15) is a hyperbolic Monge–Ampère equation, whereIn fact,Furthermore, we suppose that is the third continuous differential function and f(x) is second continuous differential function; then, initial conditions satisfyin which and .
Hence, by the standard theory of hyperbolic equations (see [16]), we have local existences and uniqueness theorem (i.e., Theorem 1).

Remark 2. In fact, the derivation of (16) and Theorem 1 can also be obtained by a direct application of the arguments therein (cf. Section 1 in [1]). When establishing the local solvability for the normal preserving flow, the authors in [1] suppose the initial curve γ0 ∈ Hk(S1) and γt(0) ∈ Hk−1(S1), k > 5/2; however, we just provide that γ0 and γt(0) are smooth.

3. Systems in Riemann Invariants

This section is concerned with the reduction of (15). Let u(x, t) be a C3 solution of equation (15) in some domain . Suppose

Condition (26) means that vertical lines t = const are free. By definition, put

The functions r and s are tangents of angles of inclinations of characteristics of equation (15), and we always call them Riemann invariants of (15). Let p = ut and q = ux; then,

Theorem 3. Let u(x, t) be a C3- solution of (15). Suppose (26) and (15) are satisfied by u. Then, the set of functions (r, s, u, p, q), where r and s are obtained from (27), p = ut, and q = ux, is a C1 solution of the system of five equations (28) in .

Theorem 4. Let (r, s, u, p, q) be a C1–solution of (28) in the domain , satisfying the initial value conditionLet be a domain in which this solution is defined. Supposeholds. Then, u(x, t) be a C3 solution of (15) in the domain , and furthermore, ut = p, ux = q, and (26) holds.
Let ,Then, we reduce equation (28) to the following Cauchy problem for a quasilinear hyperbolic system in terms of Riemann invariants:where .
Throughout this paper, we shall use the following notation: the absolute value of any vector u = (u1, …, u5) is defined asand the C0 norm of a vector function u = u(t, x) = (u1(t, x), …, un(t, x)) on a bounded domain R is defined asFirstly, suppose is bounded on R and . Take a suitable positive constant Ω such thatFor the continuous differentiable vector function u(t, x), we may defineFurthermore, for any finite set of functions set or , and we can similarly define its norm ‖Γ‖, ‖Γ‖1.
Denote R(δ0) = {(t, x)∣t ∈ [0, δ0], x ∈ R}, where R is a bounded domain, and define its exterior domain . Furthermore, suppose are C0 functions defined on E(δ0), by the standard hyperbolic theory, there exists a constant 0 < δ ≤ δ0 such that the Cauchy problem of systems (28) and (29) has an unique C1 solution on R(δ).

4. The Proof of Theorem 2

In this section, we consider the life-span of the solution by the construct method of the local solution for the Cauchy problem of the quasilinear hyperbolic system. To prove this theorem, we first consider the initial value problem of the equation for the graph (x, u(x, t)) of the curve. Since the initial curve γ0 is a smooth curve, we can find open intervals Jα, α = 1, …, N, in different coordinates and smooth functions defined on Jα such that γ0 can be described as the union of the graphs , α = 1, …, N. Assume uα is the solution with initial value for each α. Note that u(x, t) ≡ uα satisfies (16).

The following is about the proof of Theorem 2.

First, we consider quasilinear hyperbolic system (28) to be linear. Take a suitable positive constant Ω1 such that

Here, Ω1 is a positive constant derived from second a priori estimate, . Suppose the following function setand for each ν ∈ ∑(δ∣Ω, Ω1), by the linear hyperbolic system,and the initial conditions ul(0, x) = ϕl(x), x ∈ R1, l = 1, …, 5, in which , i.e., and , i.e.,

Defining the function set , we get the C0 norm estimates of the set Γ:

Hence, the C0 norm of the set Γ can be bounded as follows:

Let ; bywe have

Therefore, we can bound the C0 norm of the set Γ2[ν] as follows:

In order to prove Theorem 2, we will get some priori estimates for the solutions of Cauchy problems for linear hyperbolic systems. These estimates will be useful in proving the existence and uniqueness of solutions for the initial value problem of quasilinear hyperbolic systems.

Firstly, by the first associated integral relations of initial value problem (39),

Hereafter, the argument of each function in the l-th integrand is (τ, ξ) = (τ, ωl(τ; t, x)),and means the directional derivative along the l-th characteristic curve, i.e.,

Moreover,where (δkl(x, t)) is the inverse matrix of (δlj(t, x)), and each δkl(x, t) is also a C1 function of (t, x). Byif 0 ≤ t ≤ δ, we have

This is first a priori estimate for the solution of Cauchy problem.

We are going to construct second a priori estimate for the first derivatives of the solutions of Cauchy problem. Define the function set

By the second associated integral,

On R(δ), we have

Letobviously; we have

By (54), if 0 ≤ t ≤ δ,

By Gronwall inequality,we have

By (62), we havei.e.,

By the definition of ‖u1,

Without loss generality, we assume , and we restrict to δ ∈ (0, δ0); then,

By Taylor’s expansion,

Plugging (68) into (66) and simplifying the second order term of δ,

This is second estimate for the first derivatives of the solution of the Cauchy problem.

Next, we are going to estimate the modulus of continuity of the first-order derivatives of the solution. Let ψ(t, x) be a function on the bounded domain R. The module of continuity of ψ(t, x) is defined by the following nonnegative function:

Similarly, we can define the module of continuity of a function with any number of independent variables. Also, the modulus of continuity of a vector function ψ = (ψi(t, x))(i = 1, …, n) or a set of functions Γ = {ψi} is defined as follows:

It is easy to see that the modulus of continuity possesses the properties stated in Lemmas 1 and 2.

Lemma 1. (see [15]). It holds thatf is a continuous function on ifff is Hölder continuous on with exponent α(0 < α ≤ 1) iff there exists a constant L such thatfor any natural number N and any positive constant C, we havewhere [C] denotes the integer part of C.

Lemma 2. (see [15]). Assuming that the right hand side in each of the following formulas makes sense, thenifthenif φ = φ(t), thenlettingwe havewhere the integrand ρ(η  φ(τ, t)) is the modulus of continuity of φ with respect to t (τ is regarded as a parameter).
By the ordinary equation which ξ = ωl(τ; t, x) satisfiesand the initial conditionsIt is easy to see thatHence,Since ,By (55) and (57) and Lemma 1, on R(δ)(0 ≤ δ ≤ δ0), we haveThen,whereHence, we haveFurthermore,DefineThen, it is obvious that, on R(δ)(0 ≤ δ ≤ δ0),By the second integral relations (54) and (56), we will estimate and ρ(t, η ∣ p) for 0 ≤ t ≤ δ. By Lemmas 1 and 2 and (94), for (note that here and in the following, τ (τ ≤ t) is regarded as a parameter), we getand as τ ≤ t ≤ δ, we havewhereHence, we have