Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

Nonlinear Analysis 73 (2010) 1543–1561

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Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems Yi Zhou a , Yong-Fu Yang b,∗ a

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

b

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu, China

article

info

Article history: Received 10 November 2008 Accepted 29 April 2010 MSC: 35L50 35Q72 74K05

abstract In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant {(t , x)| t ≥ 0, x ≥ 0}. Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C 1 solutions are obtained for small initial and boundary data. We also present two applications for physical models. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Quasilinear hyperbolic system Mixed initial-boundary value problem Global classical solution Linearly degenerate Weakly linearly degenerate

1. Introduction and main results Consider the following first order quasilinear strictly hyperbolic system

∂u ∂u + A(u) = 0, (1.1) ∂t ∂x where u = (u1 , . . . , un )T is the unknown vector function of (t , x) and A(u) is an n × n matrix with suitably smooth elements aij (u) (i, j = 1, . . . , n). By the definition of strict hyperbolicity, for any given u on the domain under consideration, A(u) has n distinct real eigenvalues λ1 (u), . . . , λn (u). We furthermore suppose that λ1 (0) < · · · < λm (0) < 0 < λm+1 (0) < · · · < λn (0).

(1.2)

Let li (u) = (li1 (u), . . . , lin (u)) (resp., ri (u) = (ri1 (u), . . . , rin (u)) ) be a left (resp., right) eigenvector corresponding to λi (u) (i = 1, . . . , n): T

li (u)A(u) = λi (u)li (u) (resp., A(u)ri (u) = λi (u)ri (u)).

(1.3)

We have det |lij (u)| 6= 0 (equivalently, det |rij (u)| 6= 0).

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (Y.-F. Yang).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.057

(1.4)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

Without loss of generality, we suppose that li (u)rj (u) ≡ δij

(i, j = 1, . . . , n),

(1.5)

ri (u) ri (u) ≡ 1 (i = 1, . . . , n), T

(1.6)

where δij stands for Kronecker’s symbol. All λi (u), lij (u) and rij (u) (i, j = 1, . . . , n) are supposed to have the same regularity as aij (u) (i, j = 1, . . . , n). For the Cauchy problem of system (1.1) with the initial data t = 0 : u = φ(x),

x ∈ R,

(1.7)

where φ(x) is a C vector function with bounded C norm, many results have been obtained for the global existence of classical solutions (see [1–4]). In particular, by means of the concept of weak linear degeneracy, for small initial data with certain decaying properties, the global existence and the blow-up phenomenon of C 1 solution to Cauchy problem (1.1) and (1.7) have been completely studied (see [5–11], also see [12–14]). In terms of two basic L1 estimates, Zhou [15] furthermore relaxed the limitations on initial data and then showed that Cauchy problem (1.1) and (1.7) for a weakly linearly degenerate and strictly hyperbolic system admits a unique global C 1 solution which also satisfies L1 stability. In order to consider the effect of nonlinear boundary conditions on the global regularity of classical solution of system (1.1), Li & Wang [16] investigated the mixed initial-boundary value problem for system (1.1) on a half-unbounded domain. The result obtained by Li & Wang indicates that the interaction of nonlinear boundary conditions with nonlinear hyperbolic waves does not cause any negative effect on the global regularity of the C 1 solution, provided that the C 1 norms of initial and boundary data both decaying at infinity are small enough. In this paper, we are going to reprove the global existence result with less restrictions on the initial and boundary data. In particular, the supreme norms of the derivatives of the initial and boundary data are not assumed to be small. Moreover, for the mixed initial-boundary value problem on a bounded domain {(t , x) | t ≥ 0, 0 ≤ x ≤ L}, the results on the global regularity can be found in [17,7,18–20]. On the domain 1

1

D := {(t , x)| t ≥ 0, x ≥ 0},

(1.8)

we consider the mixed initial-boundary value problem for system (1.1) with the initial data t =0:

u = φ(x),

x≥0

(1.9)

and the boundary condition x = 0 : vs = fs (α(t ), v1 , . . . , vm ) + hs (t )

(s = m + 1, . . . , n)

(1.10)

where

vi = li (u)u (i = 1, . . . , n),

(1.11)

hs (t ) (s = m + 1, . . . , n) are given C functions of t, and 1

α(t ) := (α1 (t ), . . . , αk (t )). Let h(t ) := (hm+1 (t ), . . . , hn (t )), and we define

|u| :=

n X

! 12 u2k

k=1

for any vector value function u = (u1 , . . . , un )T . Without loss of generality, we suppose that fs (α(t ), 0, . . . , 0) ≡ 0

(s = m + 1, . . . , n).

(1.12)

We remark that, in a neighborhood of u = 0, the boundary condition (1.10) takes the same form under any possibly different choice of left eigenvectors. To state our results precisely, we shall first recall the concept of linear degeneracy and weak linear degeneracy (see [7] or [10]) as follows. Definition 1.1. For any given u on the domain under consideration, the i-th characteristic λi (u) is called linearly degenerate in the sense of P.D. Lax, provided that

∇λi (u)ri (u) ≡ 0.

(1.13)

If all characteristics λi (u) (i = 1, . . . , n) are linearly degenerate, then system (1.1) is referred to as linearly degenerate.

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

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Definition 1.2. The i-th characteristic λi (u) is weakly linearly degenerate, if, along the i-th characteristic trajectory u = u(i) (s) passing through u = 0, defined by du

= ri (u), ds s = 0 : u = 0,

(1.14) (1.15)

we have

∇λi (u)ri (u) ≡ 0,

∀ |u| small,

(1.16)

namely,

λi (u(i) (s)) ≡ λi (0),

∀ |s| small.

(1.17)

If all characteristics λi (u) (i = 1, . . . , n) are weakly linearly degenerate, then the system (1.1) is called weakly linearly degenerate. Our main results can be summarized as follows: Theorem 1.1. Suppose that system (1.1) is strictly hyperbolic and linearly degenerate. Suppose furthermore that A(u) ∈ C 2 in a neighborhood of u = 0, φ, α, fs and hs (s = m + 1, . . . , n) ∈ C 1 . Suppose finally that the conditions of C 1 compatibility are satisfied at point (0, 0) and assumption (1.2) holds. Let M = max{sup |φ 0 (x)|, sup |α 0 (t )|, sup |h0 (t )|}. x≥0

t ≥0

(1.18)

t ≥0

Then there exists a positive constant ε independent of M such that mixed initial-boundary value problem (1.1) and (1.9), (1.10) admits a unique global C 1 solution u = u(t , x) on the domain D, provided that +∞

Z

|φ 0 (x)| dx ≤ ε, 0

lim φ(x) = 0

(1.19)

x→+∞

and +∞

Z


|α 0 (t )| + |h0 (t )| dt ≤ ε.

(1.20)

0

R +∞

Remark 1.1. The conclusion in Theorem 1.1 is still valid if we remove the hypotheses limx→+∞ φ(x) = 0. Actually, 0 |φ 0 (x)| dx ≤ ε implies that the limit limx→+∞ φ(x) =: φ+ exists. Thus, we can replace u by u − φ+ to obtain the same conclusion. Theorem 1.2. Suppose that system (1.1) is strictly hyperbolic and weakly linearly degenerate. Suppose furthermore that A(u) ∈ C 2 in a neighborhood of u = 0, φ, α, fs and hs (s = m + 1, . . . , n) ∈ C 1 . Suppose finally that the conditions of C 1 compatibility are satisfied at point (0, 0) and assumptions (1.2) and (1.12) hold. Then there exists a positive constant ε independent of M such that mixed initial-boundary value problem (1.1) and (1.9), (1.10) admits a unique global C 1 solution u = u(t , x) on the domain D, provided that +∞

Z

|φ 0 (x)| dx ≤ ε, 0

+∞

Z

|φ(x)| dx ≤ 0

ε

(1.21)

M

and +∞

Z

(|h0 (t )| + |α 0 (t )|) dt ≤ ε, 0

+∞

Z

(|h(t )| + |α(t )|) dt ≤ 0

ε M

,

(1.22)

where M is defined by (1.18). We also remark that, for the weak solution to Cauchy problem (1.1) and (1.7) for general quasilinear hyperbolic systems, Bressan et al. [21] and Liu & Yang [22] obtained global L1 stability with respect to time t. When system (1.1) is weakly linearly degenerate and strictly hyperbolic, Zhou [15] also established global L1 stability of C 1 solution to Cauchy problem (1.1) and (1.7) with ‘‘small’’ initial data. Remark 1.2. Comparing Theorem 1.2 with the result established by Li & Wang [16], we reduce the restrictions on the initial and the boundary conditions, at the same time we need more restrictions on negative characteristics. Fortunately, the additional restrictions on negative characteristics are usually satisfied in many practical cases. This paper is organized as follows: In Section 2, we recall and generalize John’s formula on the decomposition of waves. Section 3 is devoted to two basic lemmas concerning the L1 estimate. In Sections 4 and 5, we prove Theorems 1.1 and 1.2,

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respectively. Finally, two applications for linearly degenerate and weakly linearly degenerate system are given respectively in Section 6. 2. Preliminaries Suppose that A(u) ∈ C 2 . By Lemma 2.5 in [10] (see also [7]), there exists an invertible C 3 transformation u = u(˜u) (u(0) = 0) such that in the u˜ -space, for each i = 1, . . . , n, the i-th characteristic trajectory passing through u˜ = 0 coincides with the u˜ i -axis at least for |˜ui | small, namely r˜i (˜ui ei ) ≡ ei ,

∀ |˜ui | small (i = 1, . . . , n),

(2.1)

where (i)

ei = (0, . . . , 0, 1 , 0, . . . , 0)T ∈ Rn

(i = 1, . . . , n).

Such a transformation is called a normalized transformation and the corresponding unknown variables u˜ = (˜u1 , . . . , u˜ n )T are called normalized variables or normalized coordinates. Let vi (i = 1, . . . , n) be defined by (1.11) and

wi = li (u)ux (i = 1, . . . , n).

(2.2)

By (1.5), we have u=

n X

vk rk (u),

(2.3)

k=1

ux =

n X

wk rk (u).

(2.4)

∂ ∂ + λi (u) ∂t ∂x

(2.5)

k=1

Let d di t

=

be the directional derivative with respect to time t along the i-th characteristic. We have (see [10] or [7]) dv i di t

=

n X

βijk vj wk ,

(2.6)

j, k=1

where

βijk (u) = (λk (u) − λi (u)) li (u)∇ rj (u)rk (u).

(2.7)

Hence, we have

βiji (u) ≡ 0,

∀ i, j.

(2.8)

Noting (2.4), we get n X ∂vi ∂(λi (u)vi ) + = Bijk (u)vj wk , ∂t ∂x j, k=1

(2.9)

or equivalently, d[vi (dx − λi (u)dt )] =

n X

Bijk (u)vj wk dt ∧ dx,

(2.10)

j, k=1

where Bijk (u) = βijk (u) + ∇λi (u)rk (u)δij .

(2.11)

By (2.7), we have, in a normalized coordinates, Bijj (uj ej ) ≡ 0,

∀ i 6= j.

(2.12)

Furthermore, when the system is weakly linearly degenerate, in normalized coordinates, we have Bijj (uj ej ) ≡ 0,

∀ |uj | small,

∀ i, j ∈ {1, . . . , n}.

(2.13)

On the other hand, we have dw i di t

=

n X j, k=1

γijk (u)wj wk ,

(2.14)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1547

where

γijk (u) =

1

{(λj (u) − λk (u)) li (u)∇ rk (u)rj (u) − ∇λk (u)rj (u)δik + (j|k)}, 2 in which (j|k) stands for all terms obtained by changing j and k in the previous terms. Hence, we have γijj (u) ≡ 0,

∀j 6= i.

(2.15)

(2.16)

When system (1.1) is linearly degenerate, we have

γiii (u) ≡ 0 ∀i ∈ {1, . . . , n}.

(2.17)

And when the system is weakly linearly degenerate, we have

γiii (ui ei ) ≡ 0 ∀|ui |small (i = 1, . . . , n).

(2.18)

Similar to (2.10), we can get n X ∂wi ∂(λi (u)wi ) + = Γijk (u)wj wk , ∂t ∂x j, k=1

(2.19)

or equivalently, d[wi (dx − λi (u)dt )] =

n X

Γijk (u)wj wk dt ∧ dx,

(2.20)

j, k=1

where

Γijk (u) =

1 2

(λj (u) − λk (u)) li (u)[∇ rk (u)rj (u) − ∇ rj (u)rk (u)].

(2.21)

Thus, we have

Γijj (u) ≡ 0,

∀ i, j ∈ {1, . . . , n}.

(2.22)

3. Two basic L 1 estimates In this section, we present two basic L1 estimates (see [15] or [23,24]). Lemma 3.1. Let ϕ = ϕ(t , x) ∈ C 1 satisfy

ϕt + (λ(t , x)ϕ)x = F , 0 < t ≤ T , x ≥ 0, t = 0 : ϕ = ϕ0 , x ≥ 0,

(3.1) (3.2)

where λ ∈ C 1 . Then for any t ∈ [0, T ], (i) if λ ≥ 0, we have

kϕ(t , ·)kL1 (R+ ) ≤ kϕ0 kL1 (R+ ) +

T

Z 0

kF (t , ·)kL1 (R+ ) dt +

T

Z

λ(t , 0)|ϕ(t , 0)| dt ,

(3.3)

0

(ii) if λ ≤ 0, we have kϕ(t , ·)kL1 (R+ ) + where kϕ(t , ·)kL1 (R+ ) =

T

Z 0

(−λ(t , 0))|ϕ(t , 0)| dt ≤ kϕ0 kL1 (R+ ) +

R +∞ 0

T

Z 0

kF (t , ·)kL1 (R+ ) dt ,

(3.4)

|ϕ(t , x)| dx.

Proof. We only prove case (i) and case (ii) can be accomplished in the same manner. To estimate kϕ0 kL1 (R+ ) , we need only to estimate l

Z

|ϕ(t , x)| dx

(3.5)

0

for any given l > 0 and then let l → +∞. From point (T , L), we draw a C 1 characteristic curve x = xr (t ) such that dxr (t ) dt

= λ(t , xr (t )),

t ≤ T,

t = T : x r = L.

(3.6) (3.7)

Choose L sufficiently large so that 0 < l ≤ xr (t ),

0 ≤ t ≤ T.

(3.8)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

Multiplying (3.1) by sgn(ϕ), where sgn(f ) is the sign function of function f , we have

|ϕ|t + (λ(t , x)|ϕ|)x = sgn(ϕ)F .

(3.9)

Thus xr (t )

Z

d dt

xr ( t )

∂ |ϕ(t , x)| dx + x0r (t )|ϕ(t , xr (t ))| ∂t 0 Z xr (t ) Z xr ( t ) (λ(t , x)|ϕ(t , x)|)x dx + x0r (t )|ϕ(t , xr (t ))| sgn(ϕ)F dx − = Z

|ϕ(t , x)| dx =

0

0

0 xr ( t )

Z =

sgn(ϕ)F dx − (λ(t , xr (t )) − x0r (t ))|ϕ(t , xr (t ))| + λ(t , 0)|ϕ(t , 0)|

0 xr ( t )

Z ≤

|F (t , x)| dx + λ(t , 0)|ϕ(t , 0)|.

(3.10)

0

Therefore, it follows that l

Z

|ϕ(t , x)| dx ≤ 0

xr (t )

Z

|ϕ(t , x)| dx

0 T

Z ≤ kϕ0 kL1 (R+ ) +

0

kF (t , ·)kL1 (R+ ) dt +

The desired estimate follows by letting l → +∞.

T

Z

λ(t , 0)|ϕ(t , 0)| dt .

(3.11)

0



Lemma 3.2. Let ϕ = ϕ(t , x) and ψ = ψ(t , x) be C functions satisfying 1

ϕt + (λ(t , x)ϕ)x = F , 0 < t ≤ T , x ≥ 0, t = 0 : ϕ = ϕ0 (x), x ≥ 0

(3.12)

ψt + (µ(t , x)ψ)x = G,

(3.14)

(3.13)

and t = 0 : ψ = ψ0 (x),

0 < t ≤ T , x ≥ 0,

x ≥ 0,

(3.15)

respectively, where λ, µ ∈ C such that there exists a positive constant δ0 independent of T 1

µ(t , x) − λ(t , x) ≥ δ0 ,

0 ≤ t ≤ T , x ≥ 0.

(3.16)

Then for any t ∈ [0, T ], (i) if λ ≥ 0 and µ > 0, we have T

Z

δ0

0

+∞

Z 0

 Z |ϕ(t , x)| · |ψ(t , x)| dxdt ≤ 4 kϕ0 kL1 (R+ ) +

T

kF (t , ·)kL1 (R+ ) dt +

0

 Z × kψ0 kL1 (R+ ) +

λ(t , 0)|ϕ(t , 0)| dt



0

T

kG(t , ·)kL1 (R+ ) dt +

0

T

Z

T

Z



µ(t , 0)|ψ(t , 0)| dt , (3.17) 0

(ii) if λ ≤ 0 and µ ≥ 0, we have Z T Z +∞ δ0 |ϕ(t , x)| · |ψ(t , x)| dxdt 0

0



T

Z

≤ 4 kϕ0 kL1 (R+ ) +

0

 Z kF (t , ·)kL1 (R+ ) dt kψ0 kL1 (R+ ) +

T

kG(t , ·)kL1 (R+ ) dt +

0

T

Z



µ(t , 0)|ψ(t , 0)| dt , (3.18) 0

(iii) if λ < 0 and µ ≤ 0, we have Z T Z +∞ δ0 |ϕ(t , x)| · |ψ(t , x)| dxdt 0

0



T

Z

≤ 4 kϕ0 kL1 (R+ ) +

0

 Z kF (t , ·)kL1 (R+ ) dt kψ0 kL1 (R+ ) +

0

T



kG(t , ·)kL1 (R+ ) dt .

Proof. We only prove case (i). Case (ii) and (iii) can be treated in the same manner. To estimate T

Z 0

Z 0

+∞

|ϕ(t , x) k ψ(t , x)| dxdt ,

(3.19)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1549

it suffices to estimate T

Z 0

l

Z

|ϕ(t , x) k ψ(t , x)| dxdt

(3.20)

0

for any given l > 0 and then let l → +∞. Similar to Lemma 3.1, from point (T , L), we draw a C 1 characteristic curve x = xr (t ) satisfying (3.6)–(3.8). Introduce ‘‘continuous Glimm’s functional’’ Q (t ) =

ZZ 0
|ψ(t , x) k ϕ(t , y)| dxdy.

(3.21)

Then, it is easy to see that dQ (t ) dt

ZZ Z xr (t ) ∂ |ψ(t , x)| dx + = x0r (t )|ϕ(t , xr (t ))| |ψ(t , x)| · |ϕ(t , y)| dxdy 0
xr (t )

Z =−

Z0Z +

0
Z

0
Z ≤ −δ0

0
|ψ(t , x) k ϕ(t , x)| dx + µ(t , 0)|ψ(t , 0)|

xr (t )

+

xr (t )

Z

0

Z

xr (t )

(µ(t , x) − λ(t , x))|ψ(t , x)||ϕ(t , x)| dx + µ(t , 0)|ψ(t , 0)| |ϕ(t , y)| dy 0 ZZ sgn(ψ)G(t , x)|ϕ(t , y)| dxdy + sgn(ϕ)|ψ(t , x)|F (t , x) dxdy |ϕ(t , x)| dx

0

|G(t , x)| dx

0

xr (t )

Z

|ϕ(t , x)| dx +

0

xr ( t )

Z

|F (t , x)| dx

0

xr (t )

Z

|ψ(t , x)| dx.

(3.22)

0

It then follows from Lemma 3.1 that dQ (t )

xr (t )

|ψ(t , x) k ϕ(t , x)| dx  Z T Z ≤ µ(t , 0)|ψ(t , 0)| kϕ0 kL1 (R+ ) + kF (t , ·)kL1 (R+ ) dt +

dt

+ δ0

Z

0

0

xr (t )

Z + 0

xr (t )

Z + 0



|G(t , x)| dx kϕ0 kL1 (R+ ) +

λ(t , 0)|ϕ(t , 0)| dt

0

0



0

T

Z

 Z |F (t , x)| dx kψ0 kL1 (R+ ) +

T

kF (t , ·)kL1 (R+ ) dt +

Z

kG(t , ·)kL1 (R+ ) dt +

Z

T

T

λ(t , 0)|ϕ(t , 0)| dt



0 T

 µ(t , 0)|ψ(t , 0)| dt .

(3.23)

0

Integrating the inequality above with respect to time t and noting that Q (0) ≤ kϕ0 kL1 (R+ ) kψ0 kL1 (R+ ) , we obtain the desired conclusion.

(3.24) 

4. Proof of Theorem 1.1 By the existence and uniqueness of local C 1 solution to the mixed initial-boundary value problem (1.1) and (1.9), (1.10), in order to prove Theorem 1.1, it suffices to establish a uniform a priori estimate on the C 0 norm of u and ∂∂ ux on the existence domain of C 1 solution u = u(t , x) (see [25]). By finite propagation speed of waves and (1.19), we have lim u(t , x) = 0,

x→+∞

∀ t ∈ [0, T ].

(4.1)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

By (1.2), there exist positive constants δ and δ0 such that

λj+1 (u) − λj (¯u) ≥ δ0 ,

∀ |u|, |¯u| ≤ δ (j = 1, . . . , n − 1).

(4.2)

For the time being it is supposed that on the existence domain of C 1 solution u = u(t , x) to mixed initial-boundary value problem (1.1) and (1.9), (1.10), we have

|u(t , x)| ≤ δ.

(4.3)

At the end of the proof of Lemma 4.1, we shall explain that this hypothesis is reasonable. Thus, in order to prove Theorem 1.1, we only need to establish a uniform a priori estimate on the supreme norm of u and w = (w1 , . . . , wn )T , which is defined by (2.2), on any given time interval [0, T ]. Thus, it follows that u(t , x) = −

+∞

Z

uy (t , y) dy.

(4.4)

x

Let T

Z

Wb (T ) = max

r =1,...,m

(−λr (u(t , 0)))|wr (t , 0)| dt ,

(4.5)

0

U∞ (T ) = sup sup |u(t , x)|,

(4.6)

0≤t ≤T x≥0

W∞ (T ) = sup sup |w(t , x)|,

(4.7)

0≤t ≤T x≥0

W1 (T ) = sup 0≤t ≤T

+∞

Z

|w(t , x)| dx,

(4.8)

0

e1 (T ) = max max sup W i=1,...,n j6=i

Cj

Z

|wi | dt ,

(4.9)

Cj

where Cj stands for any given j-th characteristic on the domain D(T ) := {(t , x)| 0 ≤ t ≤ T , x ≥ 0}.

(4.10)

Combining (4.4) with (4.8), we get

|u(t , x)| ≤

+∞

Z

|ux (t , x)| dx ≤ CW1 (T ).

(4.11)

0

Here and hereafter, C will denote a generic positive constant independent of ε and T , and the meaning of C may change from line to line. Lemma 4.1. There exists a positive constant C independent of ε , T and M such that

e1 (T ) ≤ C ε, Wb (T ), W1 (T ), W

(4.12)

W∞ (T ) ≤ CM .

(4.13)

Proof. Differentiating the boundary conditions (1.10) with respect to t yields x=0:

m X ∂vs ∂vr ∂ fs = (α(t ), v1 , . . . , vm ) + ∂t ∂v ∂t r r =1



∂ fs ∂α



(α(t ), v1 , . . . , vm )(α 0 (t ))T + h0s (t )

(s = m + 1, . . . , n).

(4.14)

Noting (1.1) and (2.4) it is easy to see that n X ∂vi ∂ = aik (u)wk (i = 1, . . . , n), (li (u) u) = −λi (u)wi + ∂t ∂t k=1

(4.15)

aik (u) = −λk (u)rkT (u)∇ li (u)u.

(4.16)

where

Then, noting (1.2) and (4.3), for δ > 0 small enough, by (4.14) and (4.15) we have

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

x = 0 : ws =

m X

fsr (α(t ), u)wr +

r =1

k X

n X

f¯si (α(t ), u)αi0 (t ) +

i =1

1551

f˜ss¯ (α(t ), u)h0s¯ (t )

s¯=m+1

(s = m + 1, . . . , n),

(4.17)

where fsr , f¯si and f˜ss¯ are all continuous functions with respect to their arguments. We introduce QW (T ) =

n XZ X

T

|wi (t , x)| · |wj (t , x)| dxdt

(4.18)

0

0

j=1 i6=j

+∞

Z

and let J1 := {1, . . . , m}, J2 := {m + 1, . . . , n},

(4.19)

then we get QW (T ) =

nX X

+

i∈J1 j∈J1 j6=i

XX

+

i∈J1 j∈J2

XX

+

i∈J2 j∈J1

T

X Xo Z i∈J2 j∈J2 j6=i

0

+∞

Z

|wi (t , x)| · |wj (t , x)| dxdt . 0

We first estimate QW (T ). (i) For i, j ∈ J1 and i 6= j, noting (2.22), it follows from (2.19) and Lemma 3.2 that T

Z

+∞

Z

0



|wi (t , x)| · |wj (t , x)| dxdt ≤ C W1 (0) +

T

Z

0

kG(t , ·)kL1 (R+ ) dt

0

2

≤ C (W1 (0) + QW (T ))2 ,

(4.20)

where G = (G1 , . . . , Gn ) and T

Gi =

n X

Γijk (u)wj wk .

(4.21)

j, k=1

(ii) For i ∈ J1 , j ∈ J2 , it also follows from (2.19) and Lemma 3.2 that T

Z 0

+∞

Z

T

 Z |wi (t , x)| · |wj (t , x)| dxdt ≤ C W1 (0) +

0

0

  Z kGi (t , ·)kL1 (R+ ) dt × W1 (0) +

T

Z

T 0

kGj (t , ·)kL1 (R+ ) dt



λj (u(t , 0))|wj (t , 0)| dt .

+

(4.22)

0

By (4.17), noting (4.3), we have

Z

T

λj (u(t , 0))|wj (t , 0)| dt Z T m k n X X X 0 0 ¯ ˜ = λj (u(t , 0)) fjr (α(t ), u(t , 0))wr (t , 0) + fji (α(t ), u(t , 0))αi (t ) + fjs¯ (α(t ), u(t , 0))hs¯ (t ) dt 0 r =1 i =1 s¯=m+1 Z m T X λj (u(t , 0)) ≤ |fjr (α(t ), u(t , 0))| (−λr (u(t , 0))|wr (t , 0)|) dt −λ r (u(t , 0)) r =1 0 ! Z T k n X X 0 0 ¯ ˜ + λj (u(t , 0)) |fji (α(t ), u(t , 0))| · |αi (t )| + |fjs¯ (α(t ), u(t , 0))| · |hs¯ (t )| dt

0

0

i=1

≤ CWb (T ) + C

s¯=m+1

T

Z


|α 0 (t )| + |h0 (t )| dt .

(4.23)

0

Thus, combining Lemma 3.1 and (2.22) gives Wb (T ) ≤ C



W1 (0) +

T

Z 0

kG(t , ·)kL1 (R+ ) dt

≤ C (W1 (0) + QW (T )) .

 (4.24)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

By (4.23)–(4.24), noting (1.20) and (2.22), we conclude from (4.22) that T

Z

+∞

Z

|wi (t , x)| · |wj (t , x)| dxdt ≤ C (ε + W1 (0) + QW (T ))2 .

(4.25)

0

0

(iii) If either i ∈ J2 , j ∈ J1 or i ∈ J2 , j ∈ J2 and i 6= j, then a similar procedure in case (ii) yields T

Z 0

+∞

Z

|wi (t , x)| · |wj (t , x)| dxdt ≤ C (ε + W1 (0) + QW (T ))2 .

(4.26)

0

By (4.20) and (4.25), (4.26), it follows from (4.18) that QW (T ) ≤ C (ε + W1 (0) + QW (T ))2 .

(4.27)

From (1.19), it is easy to see that W1 (0) ≤ C ε.

(4.28)

Thus, we have QW (T ) ≤ C ε 2

(4.29)

and then Wb (T ) ≤ C ε.

(4.30)

We next estimate W1 (T ). (i) For i ∈ J1 , by Lemma 3.1 and (2.22), we have +∞

Z

|wi (t , x)| dx ≤ W1 (0) +

T

Z

0

kG(t , ·)kL1 (R+ ) dt ≤ W1 (0) + CQW (T ).

0

(4.31)

(ii) For i ∈ J2 , it follows from Lemma 3.1 that +∞

Z

|wi (t , x)| dx ≤ W1 (0) + 0

T

Z 0

kG(t , ·)kL1 (R+ ) dt +

T

Z

λi (u(t , 0))|wi (t , 0)| dt .

(4.32)

0

Similar to (4.23), noting (1.20) and (4.30), we get T

Z

λi (u(t , 0))|wi (t , 0)| dt ≤ C ε.

(4.33)

0

For i ∈ J2 , we then have +∞

Z

|wi (t , x)| dx ≤ C (ε + W1 (0) + QW (T )) .

(4.34)

0

Thus, by (4.29)–(4.31) and (4.34), it follows that W1 (T ) ≤ C (ε + W1 (0) + QW (T )) ≤ C ε.

(4.35)

e1 (T ). We now estimate W

To this end, we need to estimate

Z

|wi | dt (i 6= j). Cj

For any (t , x) ∈ D(T ), say, point A. (i) For i ∈ J1 , there are only two possibilities: (a) The j-th characteristic passing through point A intersects the x-axis at point B (0, αj ), and the i-th characteristic passing through point A intersects the x-axis at point P (0, αi ). One can rewrite (2.20) as d (|wi (t , x)|(dx − λi (u)dt )) = sgn(wi )Gi dt ∧ dx.

(4.36)

Using Stokes formula for (4.36) on the domain ABP, by (4.29), we have

Z Z ZZ αj |wi (0, x)| dx + |Gi | dxdt |wi (t , x)|(λj (u) − λi (u)) dt ≤ Cj αi ABP ≤ W1 (0) + CQW (T ) ≤ C ε.

(4.37)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1553

e1 , we get j 6= i and then By the definition of W |λj (u) − λi (u)| ≥ δ0 .

(4.38)

Thus, it follows that

Z

|wi (t , x)| dt ≤ C ε.

(4.39)

Cj

(b) The j-th characteristic passing through point A intersects t-axis at point B (βj , 0), and the i-th characteristic passing through point A intersects x-axis at point P (0, αi ). We denote the origin (0, 0) by point O. Using Stokes formula for (4.36) on the domain ABOP and noting Lemma 3.1, we have

Z Z ZZ Z βj αi |wi (0, x)| dx + |Gi | dxdt + (−λi (u(t , 0))|wi (t , 0)|) dt |wi (t , x)|(λj (u) − λi (u)) dt ≤ Cj 0 0 ABOP ≤ C (W1 (0) + QW (T )) ≤ C ε.

(4.40)

Thus, it follows from (4.38) that

Z

|wi (t , x)| dt ≤ C ε.

(4.41)

Cj

(ii) For i ∈ J2 , there are four possibilities: (a) The j-th characteristic passing through point A intersects the x-axis at point B (0, αj ), and the i-th characteristic passing through point A intersects x-axis at point P (0, αi ). Similar to the case (a) in (i), we integrate (4.36) on the domain ABP and use the Stokes formula to get

Z

|wi (t , x)| dt ≤ C ε.

(4.42)

Cj

(b) The j-th characteristic passing through point A intersects the x-axis at point B (0, αj ), and the i-th characteristic passing through point A intersects the t-axis at point P (βi , 0). Integrating (4.36) on the domain ABOP, applying the Stokes formula and noting boundary conditions (4.17), (1.20) and Lemma 3.1, we have

Z Z ZZ Z βi αj |wi (0, x)| dx + |Gi | dxdt + (λi (u(t , 0))|wi (t , 0)|) dt |wi (t , x)|(λj (u) − λi (u)) dt ≤ Cj 0 ABOP 0 m Z T X ≤ C (W1 (0) + QW (T )) + λi (u(t , 0))|fir (α(t ), u(t , 0))| · |wr (t , 0)| dt r =1

λi (u(t , 0))

+

0

k

T

Z 0

X l =1

|f¯il (α(t ), u(t , 0))| · |αl (t )| + 0

n X

! |f˜is¯ (α(t ), u(t , 0))| ·

|h0s¯ (t )|

dt

s¯=m+1

≤ C ε.

(4.43)

Thus, it follows from (4.38) that

Z

|wi (t , x)| dt ≤ C ε.

(4.44)

Cj

(c) The j-th characteristic passing through point A intersects the t-axis at point B (βj , 0), and the i-th characteristic passing through point A intersects the x-axis at point P (0, αi ). Similarly, integrating (4.36) on the domain ABOP, applying the Stokes formula and noting boundary conditions (4.17), (1.20) and Lemma 3.1, we have

Z

|wi (t , x)| dt ≤ C ε.

(4.45)

Cj

(d) The j-th characteristic passing through point A intersects the t-axis at point B (βj , 0), and the i-th characteristic passing through point A intersects the t-axis at point P (βi , 0). Integrating (4.36) on the domain ABP and noting the Stokes formula yields

Z ZZ Z β j |Gi | dxdt + (λi (u(t , 0))|wi (t , 0)|) dt |wi (t , x)|(λj (u) − λi (u)) dt ≤ Cj ABP βi

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

≤ C (ε + W1 (0) + QW (T )) ≤ C ε.

(4.46)

Hence, for i, j ∈ {1, . . . , n} and i 6= j, we have

Z

|wi | dt ≤ C ε

(4.47)

Cj

and then

e1 (T ) ≤ C ε. W

(4.48)

Finally, we estimate W∞ (T ). (i) For i ∈ J1 and for any given point (t , x) ∈ D(T ), the i-th characteristic passing through (t , x) must intersect the x-axis at point (0, αi ). By (2.14), we get

Z n X |wi (t , x)| ≤ W∞ (0) + γijk (u)wj wk . Ci j, k=1

(4.49)

It follows from (2.16) and (2.17) that

Z n X e1 (T ). γijk (u)wj wk ≤ CW∞ (T )W Ci j, k=1

(4.50)

Therefore, for i ∈ J1 , we have

kwi (t , ·)kC 0 := sup |wi (t , x)| ≤ W∞ (0) + C ε W∞ (T ).

(4.51)

x≥0

(ii) For i ∈ J2 and for any given point (t , x) ∈ D(T ), there are still two possibilities: (a) The i-th characteristic passing through point (t , x) intersects the x-axis at point (0, αi ), then a similar procedure in (i) above yields

kwi (t , ·)kC 0 ≤ W∞ (0) + C ε W∞ (T ).

(4.52)

(b) The i-th characteristic passing through point (t , x) intersects t-axis at point (βi , 0). By (1.18) and (4.17), we conclude from (4.50) and (4.51) that

Z n X γijk (u)wj wk |wi (t , x)| ≤ |wi (βi , 0)| + Ci j, k=1 m k X X fir (α(βi ), u(βi , 0))wr (βi , 0) + f¯il (α(βi ), u(βi , 0))αl0 (βi ) ≤ r =1 l =1 n X 0 e1 (T ) + f˜is¯ (α(βi ), u(βi , 0))h¯i (βi ) + CW∞ (T )W s¯=m+1 ≤ C (M + W∞ (0) + ε W∞ (T )).

(4.53)

Thus, by (4.51)–(4.53), we have

kwi (t , ·)kC 0 ≤ C (M + ε W∞ (T )) (i = 1, . . . , n). As long as ε is sufficiently small, we get the desired (4.13), which completes the proof of Lemma 4.1.

(4.54) 

By (1.19) and (4.4), it follows from Lemma 4.1 that U∞ (T ) ≤ CW1 (T ) ≤ C ε.

(4.55)

Taking ε as sufficiently small, we get U∞ (T ) ≤

1 2

δ,

so the hypothesis (4.3) is reasonable. Theorem 1.1 is a direct consequence of Lemma 4.1.

(4.56)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1555

5. Proof of Theorem 1.2 In Section 4, we have shown that, under the assumptions of Theorem 1.1, the mixed initial-boundary value problem (1.1) and (1.9), (1.10) for a linearly degenerate and strictly hyperbolic system admits a unique global C 1 solution u = u(t , x) for all t ≥ 0. In this section, we will prove the similar result for a weakly linearly degenerate system. In order to consider the mixed initial-boundary value problem in normalized coordinates, we need the following lemma Lemma 5.1 ([16]). The boundary condition (1.10) keeps the same form under any given smooth invertible transformation u = u(˜u) (u(0) = 0). By Lemma 2.5 in [10], there exists a normalized transformation. Without loss of generality, we assume that u =

(u1 , . . . , un )T are already normalized variables.

Similar to that in Section 4, in order to prove Theorem 1.2, we only need to establish a uniform a priori estimate on the supreme norm of u and ∂∂ ux on the existence domain of C 1 solution u = u(t , x). In this section, suppose also that (4.2) and (4.3) hold. More precisely, it suffices to establish a priori estimate on the supreme norm of v = (v1 , . . . , vn )T , which is defined by (1.11), and w on any given time interval [0, T ]. Let T

Z

Vb (T ) = max

r =1,...,m

(−λr (u(t , 0)))|vr (t , 0)| dt ,

(5.1)

0

V∞ (T ) = sup sup |v(t , x)|,

(5.2)

0≤t ≤T x≥0

+∞

Z

V1 (T ) = sup 0≤t ≤T

|v(t , x)| dx,

(5.3)

0

e V1 (T ) = max max sup i=1,...,n j6=i

Z

Cj

|vi | dt ,

(5.4)

Cj

where v = (v1 , . . . , vn )T and Cj stands for any given j-th characteristic on the domain D(T ). Noting (1.11), (2.2)–(2.4) and (4.3), VR∞ (T ) is obviously equivalent to U∞ (T ), in which U∞ (T ) is defined by (4.6). R +∞ +∞ By 0 |φ 0 (x)| dx < +∞ and 0 |φ(x)| dx < +∞, we also conclude that lim u(t , x) = 0,

∀ t ∈ [0, T ].

x→+∞

(5.5)

Hence, it follows from (4.4) that

|u(t , x)| ≤

+∞

Z

|ux (t , x)| dx ≤ CW1 (T ).

(5.6)

0

Lemma 5.2. There exists a positive constant C independent of ε , T and M such that

e1 (T ) ≤ C ε, Wb (T ), W1 (T ), W Vb (T ), V1 (T ), e V1 (T ) ≤ C

ε

M

(5.7)

,

(5.8)

W∞ (T ) ≤ CM ,

(5.9)

e1 (T ) are defined by (4.5)–(4.9), respectively. where Wb (T ), W∞ (T ), W1 (T ) and W Proof. Analogous to the proof of Lemma 4.1, we furthermore introduce QV (T ) =

n XZ X j=1 i6=j

T 0

+∞

Z

|vi (t , x)| · |wj (t , x)| dxdt .

(5.10)

0

Completely repeating the procedure in the proof of Lemma 4.1 gives QW (T ) ≤ C ε 2 ,

Wb (T ) ≤ C ε,

(5.11)

where QW (T ) and is defined by (4.18). We now estimate QV (T ). (i) For i, j ∈ J1 and i 6= j, by (2.9) and (2.19), it follows from Lemma 3.2 that T

Z 0

+∞

Z 0

 Z |vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) + 0

T

kF (t , ·)kL1 (R+ ) dt 

≤ C (W1 (0) + QW (T )) V1 (0) +



W1 (0) +

0 T

Z 0

T

Z

kG(t , ·)kL1 (R+ ) dt





kF (t , ·)kL1 (R+ ) dt ,

(5.12)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

where G is defined by (4.21) and F = (F1 , . . . , Fn )T with Fi =

n X

Bijk (u)vj wk .

(5.13)

j, k=1

(ii) For i ∈ J1 , j ∈ J2 , by (2.9) and (2.19), we similarly conclude from Lemma 3.2 that T

Z 0

Z

+∞



|vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) +

T

Z

kFi (t , ·)kL1 (R+ ) dt

0

0

 ×

W1 (0) +



T

Z

kGj (t , ·)kL1 (R+ ) dt +

0

t

Z

 λj (u(t , 0))|wj (t , 0)| dt .

(5.14)

0

Noting (4.23) and (4.24), we see that T

Z

Z

+∞

 kF (t , ·)kL1 (R+ ) dt (W1 (0) + QW (T ) + ε) .

0

0

0

T

 Z |vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) +

(5.15)

(iii) For i ∈ J2 , j ∈ J1 , we have T

Z 0

Z

+∞

T

 Z |vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) +

0

kFi (t , ·)kL1 (R+ ) dt +

0

 ×

W1 (0) +

T

Z 0

T

Z

λi (u(t , 0))|vi (t , 0)| dt



0



kGj (t , ·)kL1 (R+ ) dt .

(5.16)

Noting (1.10), (1.12) and (1.22), it then follows from (4.3) that T

Z

λi (u(t , 0))|vi (t , 0)| dt =

Z

T

λi (u(t , 0)) |fi (α(t ), v1 (u(t , 0)), . . . , vm (u(t , 0))) + hi (t )| dt X Z T λi (u(t , 0)) ∂ fi ε ≤ ∂v (−λr (u(t , 0))|vr (u(t , 0))|) dt + C M −λ ( u ( t , 0 )) r r 0 r =1  ε , ≤ C Vb (T ) +

0

0 m

M

(5.17)

From Lemma 3.1, we get Vb (T ) ≤ C (V1 (0)) +

T

Z 0

kF (t , ·)kL1 (R+ ) dt

(5.18)

and T

Z 0

Z

+∞

T

 Z |vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) +

0

kF (t , ·)kL1 (R+ ) dt +

0

ε



M

(W1 (0) + QW (T )) .

(5.19)

(W1 (0) + QW (T ) + ε) .

(5.20)

(iv) For i, j ∈ J2 and i 6= j, a similar argument yields T

Z 0

Z

+∞

T

 Z |vi (t , x)| · |wj (t , x)| dxdt ≤ C V1 (0) +

0

0

kF (t , ·)kL1 (R+ ) dt +

ε M



Combining four cases above together and noting (4.28), (5.11) and (1.21), we then have QV (T ) ≤ C ε



ε

T

Z

M

+ 0



kF (t , ·)kL1 (R+ ) dt .

(5.21)

Noting (2.13), one can use Hadamard’s formula to get Bijj (u) = Bijj (u) − Bijj (uj ej )

=

X

1

Z uh 0

h6=j

∂ Bijj (su1 , . . . , suj−1 , uj , suj+1 , . . . , un ) ds. ∂ uh

(5.22)

Thus

|Bijj (u)| ≤ C

X h6=j

|uh |,

∀ i, j ∈ {1, . . . , n}.

(5.23)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1557

For any h 6= j, uh = u · e h =

n X

vk rk (u) · eh = vj rj (u) · eh +

k=1

X

vk rk (u) · eh .

(5.24)

k6=j

Noting (2.1), we have

|rj (uj ej ) · eh | ≡ 0. ∀ j 6= h.

(5.25)

Hadamard’s formula gives

|rj (u) · eh | = C

X

|uk | (∀ j 6= h).

(5.26)

k6=j

It follows from (5.24) that

X

| uh | ≤ C

X

h6=j

|vh | + C |v|

h6=j

X

|uh | ≤ C

h6=j

X

|vh | + C δ

X

h6=j

|uh |.

(5.27)

h6=j

Thus, it is easy to see that

X

| uh | ≤ C

X

h6=j

|vh |,

(5.28)

h6=j

provided δ is small enough. By (5.23) and (5.28), we get T

Z 0

kF (t , ·)kL1 (R+ ) dt ≤ CQV (T ).

(5.29)

Hence, combining (5.29) and (5.21) yields QV (T ) ≤ C

ε2

(5.30)

M

and then Vb (T ) ≤

ε M

.

(5.31)

For a weakly linearly degenerate system, one can use the same procedure as the proof of Lemma 4.1 to show that

e1 (T ) ≤ C ε. W1 (T ), W

(5.32)

e1 (T ), noting boundary conditions (1.10) and (1.12), we can also prove that Similar to the proof of W1 (T ), W V1 (T ), e V1 (T ) ≤ C

ε M

.

(5.33)

Finally, we estimate W∞ (T ). (i) For i ∈ J1 and for any given point (t , x) ∈ D(T ), the i-th characteristic passing through (t , x) must intersect the x-axis at point (0, αi ). By (2.14), we have

Z n X |wi (t , x)| ≤ W∞ (0) + γijk (u)wj wk . Ci j, k=1

(5.34)

By (2.16) and (2.18), it follows that

Z Z n X e1 (T ) γijk (u)wj wk ≤ |γiii (u)wi2 | + CW∞ (T )W C i j , k =1 Ci Z 2 e1 (T ). ≤ C W∞ (T ) |γiii (u)| + CW∞ (T )W

(5.35)

Ci

By Hadamard’s formula, we get

|γiii (u)| = |γiii (u) − γiii (ui ei )| ≤ C

X j6=i

then

| uj | ≤ C

X j6=i

|vj |,

(5.36)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

Z

|γiii (u)| ≤ C e V1 (T ).

(5.37)

Ci

Thus, for i ∈ J1 ,

kwi (t , ·)kC 0 ≤ W∞ (0) + C

ε M



2 W∞ (T ) + ε W∞ (T ) .

(5.38)

(ii) For i ∈ J2 and for any given point (t , x) ∈ D(T ), similar to the proof of Lemma 4.1, noting (5.35) and (5.37), we also conclude

kwi (t , ·)kC 0 ≤ C (M + W∞ (0)) + C

ε M



2 W∞ (T ) + ε W∞ (T ) .

Thus, combining (i) and (ii) gives the desired (5.9).

(5.39)



It follows from (5.6) that U∞ (T ) ≤ C W1 (T ) ≤ C ε.

(5.40)

Taking ε sufficiently small, we then get 1

U∞ (T ) ≤

δ,

(5.41)

2 so that the hypothesis (4.3) is reasonable as well. Obviously, Lemma 5.2 implies Theorem 1.2. 6. Applications

In this section, we give two applications of the foregoing results for a linearly degenerate system and a weakly linearly degenerate system, respectively. 6.1. The planar motion of elastic string Consider the following mixed initial-boundary value problem for the system of the planar motion of an elastic string (see [11,9])

 vx = 0 ,   ut −  T (r ) u = 0, vt − r

(6.1)

x

with the initial condition u = u˜ 0 + u0 (x),

t =0:

v = v˜ 0 + v0 (x) (x ≥ 0)

(6.2)

and the dissipative boundary condition T (r )

(α > 0 is a constant), q where u = (u1 , u2 )T , v = (v1 , v2 )T , r = |u| = u21 + u22 , T (r )(r > 1) is a C 3 function of r > 1, such that x=0:

T 0 (˜r0 ) >

r

T (˜r0 ) r˜0

u = αv

(6.3)

> 0,

(6.4)

where r˜0 = |˜u0 | > 1, u˜ 0 , v˜ 0 are constant vectors and (u0 (x)T , v0 (x)T ) ∈ C 1 . Let M = max{sup |u00 (x)|, sup |v00 (x)|}. x≥0

(6.5)

x≥0

Suppose there exists a positive constant ε > 0 such that +∞

Z

(|u00 (x)| + |v00 (x)|) dx ≤ ε, 0

+∞

Z

(|u0 (x)| + |v0 (x)|) dx ≤ 0

ε M

(6.6)

and the conditions of C 1 compatibility are supposed to be satisfied at point (0, 0). Let

  U =

u

v

.

(6.7)

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1559

  u˜

By (6.4), in a neighborhood of U0 = v˜ 0 , (6.1) is a strictly hyperbolic system with the following distinct real eigenvalues 0

r r p p T (r ) T (r ) 0 < 0 < λ3 (U ) = < λ4 (U ) = T 0 (r ) λ1 (U ) = − T (r ) < λ2 (U ) = − r

r

(6.8)

and the corresponding left eigenvectors can be taken as l1 (U ) =

p

T 0 (r )uT , uT

r l3 (U ) =

T (r ) r



r ,

l2 (U ) =

r

! w, −w ,

T (r )

! w, w ,

p

l4 (U ) =



T 0 (r )uT , −uT ,

(6.9)

where w = (−u2 , u1 ). It is easy to see that λ2 (U ) and λ3 (U ) are linearly degenerate. Moreover, λ1 (U ) and λ4 (U ) are also linearly degenerate, provided that T 00 (r ) ≡ 0,

∀ |r − r0 | small.

(6.10)

Let Vi = li (U )(U − U0 )

(i = 1, . . . , 4).

(6.11)

It is easy to see that (6.3) can be rewritten as x=0:

V3 = f3 (V1 , V2 ),

V4 = f4 (V1 , V2 ),

(6.12)

1

where f3 and f4 are C functions with respect to their arguments and f3 (0, 0) = f4 (0, 0) = 0.

(6.13)

By Theorem 1.1, we have Theorem 6.1. Suppose that (6.10) holds. There exists a constant ε > 0 small enough such that the mixed initial-boundary value problem (6.1)–(6.3) admits a unique global C 1 solution on the domain D. 6.2. The compressible elastic fluids with memory Consider the following mixed initial-boundary value problem for the system of compressible elastic fluids with memory

( ρt + vρx + vx ρ = 0, ρ(vt + vvx ) + p(ρ)x = (ρ W 0 (F )F )x , Ft + v Fx − vx F = 0,

(6.14)

with the initial condition t =0:

ρ = ρ0 + ρ1 (x) (ρ0 > 0), v = v0 + v1 (x) (v0 > 0), F = 1 + F1 (x)

(6.15)

and the boundary condition x=0:

v = v0 + v2 (t ),

F = 1 + F2 (t ),

(6.16)

where ρ is the density, v is the velocity, p is the pressure, W (F ) is the strain energy function and F is deformation tensor, ρ1 (x), v1 (x), F1 (x) ∈ C 1 [0, +∞) and v2 (t ), F2 (t ) ∈ C 1 [0, ∞). Let

o

n

M = max sup{|ρ10 (x)| + |v10 (x)| + |F10 (x)|}, sup{|v20 (t )| + |F20 (t )|} .

(6.17)

t ≥0

x≥0

Suppose that there exists a constant ε > 0 such that +∞

Z


|ρ10 (x)| + |v10 (x)| + |F10 (x)| dx ≤ ε,

(6.18)

0

+∞

Z

(|ρ1 (x)| + |v1 (x)| + |F1 (x)|) dx ≤ 0

ε

(6.19)

M

and +∞

Z


|v20 (t )| + |F20 (t )| dt ≤ ε, 0

+∞

Z

(|v2 (t )| + |F2 (t )|) dt ≤ 0

ε M

.

(6.20)

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Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

Suppose furthermore that the conditions of C 1 compatibility hold at point (0, 0) and

v0 <

p

p0 (ρ0 ) + W 00 (1)

(6.21)

holds as well. Let

! ρ U = v ,

(6.22)

F by (6.21), in a neighborhood of U 0 = (ρ0 , v0 , 1)T , (6.14) is a strictly hyperbolic system with three distinct real eigenvalues

p

λ1 = v −

p0 (ρ) + W 00 (F )F 2 < 0 < λ2 = v < λ3 = v +

p

p0 (ρ) + W 00 (F )F 2

(6.23)

and the corresponding left eigenvectors can be taken as W 0 (F )F − p0 (ρ)

l1 (U ) =

W 00 (F )F + W 0 (F )

p , 1, p ρ p0 (ρ) + W 00 (F )F 2 p0 (ρ) + W 00 (F )F 2   F l2 (U ) = , 0, 1 , ρ

! ,

W 00 (F )F + W 0 (F )

W 0 (F )F − p0 (ρ)

(6.24)

!

, 1, − p − p ρ p0 (ρ) + W 00 (F )F 2 p0 (ρ) + W 00 (F )F 2

l3 (U ) =

.

It is easy to see that λ2 (U ) is linearly degenerate, then weakly linearly degenerate. On the one hand, the first characteristic trajectory U = U (1) (s) passing through U 0 is defined by

( dU

= r1 (U ), ds s = 0 : U = U 0,

where r1 (U ) =

 −√

(6.25)

ρ

p0 (ρ)+W 00 (F )F

, 1, √ 2

T

F

p0 (ρ)+W 00 (F )F 2

. By the first equation and the third one in system (6.25), we

immediately get d(ρ F ) ds

= 0,

(6.26)

and noting that s = 0 : ρ = ρ0 , F = 1, thus, ρ F ≡ ρ0 always holds along the first characteristic trajectory U = U (1) (s) passing through U 0 . On the other hand, it is easy to see that λ1 (U ) is linearly degenerate if and only if

ρ p00 (ρ) + 2p0 (ρ) − W 000 (F )F 3 ≡ 0.

(6.27)

Thus, by definition, λ1 (U ) is weakly linearly degenerate if and only if

ρ p00 (ρ) + 2p0 (ρ) − W 000 (ρ0 ρ −1 )ρ03 ρ −3 ≡ 0.

(6.28)

Similarly, λ3 (U ) is weakly linearly degenerate if and only if (6.28) holds. Let Vi = li (U )(U − U 0 )

(i = 1, 2, 3).

(6.29)

By the Implicit Function Theorem, boundary condition (6.16) can be rewritten as x=0:

V2 = f2 (v2 (t ), F2 (t ), V1 ) + h2 (t ),

V3 = f3 (v2 (t ), F2 (t ), V1 ) + h3 (t ),

(6.30)

where f2 (v2 (t ), F2 (t ), 0) = f3 (v2 (t ), F2 (t ), 0) = 0,

(6.31)

and h2 (t ), h3 (t ) ∈ C 1 [0, +∞) satisfying +∞

Z

(|h02 (t )| + |h03 (t )|) dt ≤ ε, 0

Z 0

+∞

(|h2 (t )| + |h3 (t )|) dt ≤

ε M

.

(6.32)

By Theorem 1.2, we immediately have Theorem 6.2. Suppose that (6.28) holds. There exists a constant ε > 0 so small that the mixed initial-boundary value problem (6.14)–(6.16) admits a unique global C 1 solution on the domain D.

Y. Zhou, Y.-F. Yang / Nonlinear Analysis 73 (2010) 1543–1561

1561

Acknowledgements The authors would like to thank Professor Li Ta-Tsien for his sustained support, guidance and encouragement. Y. Zhou is supported by the National Natural Science Foundation of China under grant 10728180, the 973 Project of the Ministry of Science and Technology of China, the Doctoral program foundation of the Ministry of Education of China and the ‘‘111 project’’ (B08018). The second author was in part supported by the National Natural Science Foundation of China under grant 10926162 and the Natural Science Foundation of Hohai University under grant 2009428011. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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