伝播速度の異なる非線型波動方程式系の大域解について

北海道教育大学函館校久保田幸次

この講演の内容は，横山和義氏との共同研究

Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation

Koji KUBOTA and Kazuyoshi YOKOYAMA

(to appear in Japanease J. Math., Vol. 27, No.1)

に基づいている．

Abstract. This paper is concerned with a system of nonlinear wave equations in three space dimensions

∂_{t}^{2}u^{i} − c_{i}^{2}Δu^{i} = F^{i}(u, ∂u, ∂^{2}u), i = 1, 2, ⋯, m,

where 0 < c_{1} < c_{2} < ⋯ < c_{m}. We prove the global existence of classical solutions to the system with small initial data, provided F^{i} satisfy “Null conditon”.

§1. Introduction.

This paper is a continuation of a previous paper [21] on global (in time) existence of small amplitude solutions to a system of quasilinear wave equations with different speeds of propagation. Indeed, the second author studied in it the system stated in the above abstract fot the case that F^{i} depend only on the first and second derivatives ∂u, ∂^{2}u of u, i.e., that F^{i} = F^{i}(∂u, ∂^{2}u). The purpose of present paper is to extend the results of [21] to the case where F^{i} = F^{i}(u, ∂u, ∂^{2}u) depend explicitly on u as well as ∂u and ∂^{2}u. As a result, we will refine the proof of the main theorem in Klainerman [14] which deals with the case of one speed of propagation, i.e., m = 1, in the sense that we do not need to make use of the vector fields L_{j} = x_{j}∂_{t} + t∂_{j}, j = 1, 2, 3, where ∂_{t} = ∂/∂_{t} and ∂_{j} = ∂/∂_{x_{j}}.

We consider the following Cauchy problem

□_{i}u^{i} ≡ ∂_{t}^{2}u^{i} − c_{i}^{2}Δu^{i} = F^{i}(u, ∂u, ∂^{2}u) (1.1)

in (0, ∞) × ℝ^{3}, i = 1, 2, ⋯, m,

u^{i}(0, x) = εf^{i}(x), ∂_{t}u^{i}(0, x) = εg^{i}(x) (1.2)

for x ∈ ℝ^{3}, i = 1, 2, ⋯, m,

where c_{1}, c_{2}, ⋯, c_{m} are positive constants different each other, say, 0 < c_{1} < c_{2} < ⋯ < c_{m}, f^{i} and g^{i} are fixed functions in C_{0}^{∞}(ℝ^{3}) and ε is a positive parameter which serves to measure the amplitude of the initial values. Here u^{i} = u^{i}(t, x) = (u_{1}^{i}(t, x), u_{2}^{i}(t, x), ⋯, u_{p_{i}}^{i}(t, x)) is a vector of p_{i} real valued functions of (t, x) ∈ [0, ∞) × ℝ^{3}, u = (u^{1}, u^{2}, ⋯, u^{m}) a p-vector, where p = p_{1} + p_{2} + ⋯ + p_{m}, ∂ = (∂_{t}, ∂_{x}), ∂_{t} = ∂_{0} = ∂/∂_{t}, ∂_{x} = (∂_{1}, ∂_{2}, ∂_{3}), ∂_{j} = ∂/∂_{x_{j}} for j = 1, 2, 3, Δ = ∑_{j=1}^{3} ∂_{j}^{2} and ∂^{2}u stands for second derivatives of u. For convenience we will often denote ∂u and ∂^{2}u by u′ and u′′, respectively. Moreover F^{i}(u, u′, u′′) is a vector of p_{i} real valued functions of (u, u′, u′′), belonging to C^{∞}(ℝ^{21p}), such that

(1.3.1)  F^{i}(0, 0, 0) = 0, i = 1, 2, ⋯, m

and

(1.3.2)  (F^{i})′(0, 0, 0) = 0, i = 1, 2, ⋯, m.

Here and in what follows we denote by (F^{i})′(u, u′, u′′) the first derivatives of F^{i} with respect to (u, u′, u′′) and by ∂_{u}F^{i}(u, u′, u′′) the first derivatives of F^{i}(u, u′, u′′) with respect to u, and so on.

Remark. The hypothesis (1.3.1) implies that the Cauchy problem (1.1)-(1.2) has a trivial solution u = 0, provided ε = 0. Moreover the condition (1.3.2) means that the linear part of (1.1) coincides with □_{i}. For the smoothness of F^{i} it is sufficient to assume that F^{i}(u, u′, u′′) is of class C^{∞} on a neighborhood of (u, u′, u′′) = (0, 0, 0). However such a function F^{i} can be always extended to the whole space ℝ^{21p} by multiplying a cutoff function. For the sake of simplicity of description we have therefore assumed that F^{i} ∈ C^{∞}(ℝ^{21p}).

We shall assume without loss of generality that the system (1.1) is quasilinear, i.e., that F^{i}(u, u′, u′′) is linear in u′′, hence one can write

(1.4.1)  \begin{matrix}F^{i}(u, u′, u′′) = ∑_{j=1}^{m} ∑_{a,b=0}^{3}A_{ij}^{ab}(u, u′)∂_{a}∂_{b}u^{j} + B^{i}(u, u′) \\ for i = 1, 2, ⋯, m, \end{matrix}

where A_{ij}^{ab}(u, u′) is a p_{i} × p_{j} matrix function of (u, u′) ∈ ℝ^{p} × ℝ^{4p} and u^{j} are regarded as column vectors. (See for instance Courant and Hilbert [4], chapter I, §7). We suppose that

(1.4.2)  A_{ij}^{ba}(u, u′) = A_{ij}^{ab}(u, u′) for a, b = 0, 1, 2, 3 and i, j = 1, 2, ⋯, m,

since only classical solutions are considered in this paper. Clearly, conditions (1.3) and (1.4.1) imply that

(1.4.3)  A_{ij}^{ab}(0, 0) = 0 for i, j = 1, 2, ⋯, m, a, b = 0, 1, 2, 3

and

(1.4.4)  B^{i}(0, 0) = 0, (B^{i})′(0, 0) = 0 for i = 1, 2, ⋯, m.

In order to assure the existence of local (in time) solutions, we also require that the system (1.1) can be reduced to a symmetric hyperbolic system of the first order. Namely, we assume that the p × p matrices

(A_{ij}^{ab}(u, u′); i, j = 1, 2, ⋯, m)

are (real) symmetric for a, b = 0, 1, 2, 3 where A_{ij}^{ab}(u, u′) are given by (1.4.1), i.e., that

A_{ij}^{ab}(u, u′) =^{t} A_{ij}^{ab}(u, u′) for i, j = 1, 2, ⋯, m, a, b = 0, 1, 2, 3, (1.5)

where ^{t}A stands for the transposed matrix of A.

Let F^{i,2}(u, u′, u′′) be the quadratic part of Taylor’s expansion for F^{i}(u, u′, u′′) about (u, u′, u′′) = (0, 0, 0). Then by the assumptions (1.3) and (1.4.1) one can write

F^{i}(u, u′, u′′) = F^{i,2}(u, u′, u′′) + O((|u| + |u′|)^{2}(|u| + |u′| + |u′′|)) (1.6)

for (u, u′, u′′) near (0, 0, 0).

We shall devide all terms of the homogeneous polynomial F^{i,2}(u, u′, u′′) into two group as follows.

F^{i,2}(u, ∂u, ∂^{2}u) (1.7)

= ∑_{k=1}^{m} {N_{k}^{i}(u^{k}, ∂u^{k}, ∂^{2}u^{k}) + R_{k}^{i}(u, ∂u, ∂^{2}u)}
for i = 1, 2, ⋯, m,

where N_{k}^{i}(u^{k}, ∂u^{k}, ∂^{2}u^{k}) are homogeneous polynomials in (u^{k}, ∂u^{k}, ∂^{2}u^{k}) of degree 2, and R_{k}^{i}(u, ∂u, ∂^{2}u) are homogeneous polynomials in (u, ∂u, ∂^{2}u) of degree 2 which are linear in (u^{k}, ∂u^{k}, ∂^{2}u^{k}). Then we assume that for each i, k = 1, 2, ⋯, m the polynomial N_{k}^{i}(u^{k}, ∂u^{k}, ∂^{2}u^{k}) satisfies the \underline{null condition} in the sense of Christodoulou [3]. Besides, we suppose that for each i, k = 1, 2, ⋯, m the polynomial R_{k}^{i}(u, ∂u, ∂^{2}u) does not explicitly depend on u, i.e., that

∂_{u}R_{k}^{i}(u, u′, u′′) = 0 for i, k = 1, 2, ⋯, m. (1.8)

Then we have the following.

Theorem 1. Suppose that (1.4), (1.5) and (1.8) hold. Assume that N_{k}^{i}(u^{k}, ∂u^{k}, ∂^{2}u^{k}) satisfies the null condition for each i, k = 1, 2, ⋯, m. Let k be a positive integer with k ≥ 14 and let ν be a positive number with ν < 1/2. Then there are positive constants ε_{0} and C such that for any 0 < ε ≤ ε_{0} there exists uniquely a solution u(t, x) ∈ C^{∞}([0, ∞) × ℝ^{3}) of (1.1)-(1.2) satisfying

|u^{i}(t, x)|(1 + τ + t)(1 + |τ − c_{i}t|)^{ν} + ∑_{1≤|α|≤k} |∂^{α}u^{i}(t, x)|(1 + τ) (1.9)

× (1 + |τ − c_{i}t|)^{1+ν} ≤ Cε for (t, x) ∈ [0, ∞) × ℝ^{3}, i = 1, 2, ⋯, m

and

(1 + t)^{−1}∥u(t)∥_{L^{2}} + ∑_{1≤|α|≤k+6} ∥∂^{α}u(t)∥_{L^{2}} ≤ Cε for t ≥ 0, (1.10)

where τ = |x| and

∥v(t)∥_{L^{2}} = {∫_{ℝ^{3}} |v(t, x)|^{2}dx}^{1/2} .

Here the constants ε_{0} and C depend only on k, ν, F^{i}, f^{i} and g^{i}.

References

[1] Agemi R. and Yokoyama K., The null condition and global existence of solutions to systems of wave equations with different speeds, in ‘Advances in nonlinear partial differential equations and stochastics’ (S. Kawashima and T. Yanagisawa ed.), Series on Adv. in Math. for Appl. Sci., Vol.48, 43–86, World Scientific, 1998.

[2] Asakura F., Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm, Partial Differential Equations, 11 (1986), 1459–1487.

[3] Christodoulou D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267–282.

[4] Courant R. and Hilbert D., Methods of mathematical physics, Vol.II, Interscience Publ., 1962.

[5] Hoshiga A. and Kubo H., Global small amplitude solutions of nonlinear hyperbolic system with a critical exponent under the null condition, to appear in SIAM Jour. of Math. Analysis.

[6] John F., Blow-up for solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235–268.

[7] John F., Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29–51.

[8] John F., Lower bounds for the life span of solutions of nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 36 (1983), 1–35.

[9] John F., Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math., 40 (1987), 79–109.

[10] John F., Nonlinear wave equations, Formation of singularities, Pitcher Lectures in the mathematical sciences, Lehigh University, University Lecture Series, American Math. Soci., Providence, (1990).

[11] Katayama S., Global existence for systems of nonlinear wave equations in two space dimensions, II, Publ. RIMS, Kyoto Univ., 31 (1995), 645–665.

[12] Kato T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181–205.

[13] Klainerman S., Long time behavior of solutions to nonlinear wave equations, Proceedings of the International Congress of Mathematicians, Warsaw, (1983), 1209–1215.

[14] Klainerman S., The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math., 23 (1986), Amer. Math. Soc., 293–326.

[15] Klainerman S. and Sideris T.C., On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math., 49 (1996), 307–322.

[16] Kovalyov M., Resonance-type behaviour in a system of nonlinear wave equations, J. Differential Equations, 77 (1989), 73–83.

[17] Lax P.D. and Phillips R.S., Scattering theory, Academic Press, New York and London, 1967.

[18] Li Ta-tsien and Zhou Yi, Life-span of classical solutions to fully nonlinear wave equations-II, Nonlinear Anal., 19 (1992), 833–853.

[19] Lindblad H, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445–472.

[20] Madja A., Compressible fluid flow and systems of conservation laws in several space variable, Applied Math. Sci., 53, Springer-Verlag, 1984.

[21] Yokoyama K., Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, 52 (2000), 609–632.