- Email: [email protected]
Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity
Accepted Manuscript Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity
Yun-guang Lu, Yuusuke Sugiyama
PII: DOI: Reference:
S0893-9659(18)30143-5 https://doi.org/10.1016/j.aml.2018.05.001 AML 5508
To appear in:
Applied Mathematics Letters
Received date : 23 March 2018 Revised date : 1 May 2018 Accepted date : 1 May 2018 Please cite this article as: Y. Lu, Y. Sugiyama, Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity Yun-guang Lu∗and Yuusuke Sugiyama†
Abstract We prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations vtt = c(|v|s−1 v)xx with the nonnegative initial data v0 (x) and s > 1. These results are extensions of the results in the second author’s paper [Sugiyama SIAM J. Math. Anal., 48(2016), 847-860], where the existence and the∫ nonexistence of the unique global classical solution were studied ∞ with a threshold on −∞ v1 (x)dx and the non-degeneracy condition v0 (x) ≥ c0 > 0 on the initial data.
Key Words: Global weak solutions; degenerate wave equations; viscosity method; compensated compactness Mathematics Subject Classification 2010: 35L15, 35A01, 62H12.
1
Introduction
In this paper, we study the global generalized solutions of the nonlinearly degenerate wave equation vtt = c(|v|s−1 v)xx , −∞ < x < ∞, t > 0, (1.1) with the initial data (v, vt )|t=0 = (v0 (x), v1 (x)),
−∞ < x < ∞.
(1.2)
2
where v0 (x) ≥ 0, s > 1, c = θs > 0 and θ = s+1 2 > 0 are constants. The wave equation (1.1) ∫arises in the theory of elasticity and elasto-plasticity (see Cristescu [1]). v and x u = −∞ ∂t vdx describe the strain and the particle velocity and σ(v) = c|v|s−1 v s is the stress function. The exponent s in σ(v) is determined by materials. In our case s > 1, the stress function is applied to rubbers, soils and certain metals. In the case that 0 < s < 1, work hardening materials (see p. 15 in [1]). A function v ∈ L∞ (R × [0, T ]) is called a generalized solution of the Cauchy problem (1.1)-(1.2) if for any test function ϕ ∈ C0∞ (R × [0, ∞)), ∫ T∫ ∞ ∫ ∞ s−1 vϕ(x, t)tt − c|v| vϕ(x, t)xx dxdt + v0 (x)ϕ(x, 0)t − v1 (x)ϕ(x, 0)dx = 0. (1.3) 0
∗
−∞
−∞
K.K.Chen Institute for Advanced Studies Hangzhou Normal University, P. R. CHINA [email protected] † Corresponding author. Department of Mathematics, Tokyo University of Science Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan, e-mail:[email protected]
1
If we can take T = ∞ in (1.3), we call the function v ∈ L∞ (R × R+ ) global generalized solution of (1.3) Throughout this paper, we concentrate our study on the domain of v ≥ 0, then The Riemann invariants for (1.1) are given by w = u − vθ , z = u ˜ + vθ , where u=
∫
x
−∞
vt dx, u ˜=−
∫
∞
vt dx.
(1.4)
(1.5)
x
We denote w(x, 0) and z(x, 0) by w0 (x) and z0 (x) respectively. To state our main theorems, we define a threshold Γ as ∫ ∞ Γ= v1 (x)dx + v0θ (x)|x=+∞ + v0θ (x)|x=−∞ (1.6) −∞
We have the following first main result in this paper. Theorem 1. Let v0 (x) ≥ 0 be bounded, v1 (x) ∈ L1 (R) and the limits v0θ (x)|x=±∞ exist. Moreover suppose that Γ ≥ 0 and s−1
v1 (x) ± θv0 2 (x)∂x v0 (x) ≤ 0.
(1.7)
Then the Cauchy problem (1.1) and (1.2) has a global generalized solution satisfying (1.3). The proof of Theorem 1 is almost done in [7]. So we only give a comment on the proof of Theorem 1 in Remark 5. Our main contribution of this paper is the second main theorem, which implies the nonexistence of generalized solutions satisfying v ≥ 0. Furthermore, Theorem 2 give a upper estimate of the maximal existence time of solutions, which is not given in the proof of [8]. In order to state the second main theorem, we collect some conditions for global solutions. We can easily check that the global generalized solutions of (1.1) constructed Theorem 1 satisfy that v(x, t) ≥ 0 for a.a. (x, t) ∈ R × R+
(1.8)
w(x, t) and z(x, t) are decreasing with x for a.a. t ≥ 0,
(1.9)
w, z ∈ L∞ (R × R+ ).
(1.10)
Furthermore, we can show that the following properties are also satisfied for the global solutions in Theorem 1, if we assume the additional regularity on initial data that w0 , z0 ∈ 1,1 Wloc (R): 1,1 w, z ∈ Wloc (R × R+ ) and ∂t v(t, ·) ∈ L1 (R),
(1.11)
∥wx (t)∥L1 + ∥zx (t)∥L1 ≤ ∥wx (0)∥L1 + ∥zx (0)∥L1 ,
(1.12)
∥wt (t)∥L1 + ∥zt (t)∥L1 ≤ C(∥wx (0)∥L1 + ∥zx (0)∥L1 )
(1.13)
and
for a.a. t ≥ 0. We have the following second main result in this paper. 2
1,1 (R) ∩ L∞ (R) and v1 ∈ L1 (R). Suppose that Theorem 2. Let v0 ≥ 0, w0 , z0 ∈ Wloc the limits v0θ (x)|x=±∞ exist. Furthermore, we assume that (1.7) is satisfied and Γ < 0. Then the Cauchy problem (1.1) and (1.2) has no solutions satisfying the properties (1.8)(1.13). Furthermore, solutions satisfying (1.8)-(1.13) can not exist beyond the time T ∗ = ∫M 2M ∥u0 ∥L∞ /−Γ where M is a positive constant such that −M v1 (x)dx+v0θ (M )+v0θ (−M ) < 0.
Theorems 1 and 2 extend the results in [8]. In [8], under the assumption (1.7), the second author has already obtained the threshold Γ separating the global existence of solutions and the occurrence of the degeneracy in finite time. However in [8], it is assume that initial data are sufficiently smooth and that v0 (x) ≥∃ c0 > 0 for all x ∈ R. We note that the hyperbolicity is lost when the equation degenerate. Under the assumption that solutions are nonnegative, the decreasing property of w0 and z0 ensures the absence of shock waves. So we can expect that generalized solutions do not satisfy the regularity properties (1.11)-(1.13) are not satisfied, if nonnegativity of solutions does not hold. This expectation is a motivation for Theorem 2. The main theorem in [8] can not estimate the time when the degeneracy occurs. Furthermore, non-smooth solutions are not treated in [8], since the proof of the main theorem is based on the method of characteristics. While H¨older continuous, BV or L∞ solutions has been discussed in many papers (e.g. [5, 6, 7] and Lions, Perthame, Souganidis and Tadmor [4, 3]) ∫ ∞for 1D hyperbolic conservation law. The key for the proof in [8] is the function F (t) = −∞ v(x, 0) − v(x, t)dx In the estimate for F (t), we divide the integral region (−∞, ∞) of F by three parts, using characteristic curves. However, for non-classical solutions, the characteristic curves would not be defined. Observing 0 ≤ wt (x, t) and zt (x, t) ≤ 0, we estimate F more simply.
2
Proof of Theorem 2
∫M 1,1 We note that w0 , z0 ∈ C(R) ⊂ Wloc (R) and −M v1 (x)dx + v0θ (M ) + v0θ (−M ) = w0 (M ) − z0 (−M ). Hence, since w0 and z0 are decreasing, the assumption Γ < 0 implies that lim w0 (x) ≥ w0 (x) > z0 (y) ≥ lim z( x). x→∞
x→−∞
Hence there exists a large constant M0 > 0 such that if M ≥ M0 , then w0 (−M ) > z0 (M ). Lemma 3. Suppose that the assumptions (1.8)-(1.13) are satisfied. Then we have for a.a. (x, t) ∈ R × R+ . { vt − ux = 0, (2.1) ut − c(|v|s−1 v)x = 0 and
{
wt + λ2 wx = 0, zt + λ1 zx = 0.
(2.2)
1,1 This Lemma states that for wloc solutions, the weak form of the equation can be reduced to a 2 × 2 hyperbolic system (not weak sense) under the assumptions (1.8)-(1.13). The proof is based on standard density argument. So we omit it. To state the next lemma, we put limx→±∞ v0 (x) = v± .
3
Lemma 4. Suppose that the assumptions (1.8)-(1.13) is satisfied, then we have lim v(x, t) = v± .
(2.3)
x→±∞
Proof. Since w and z are decreasing with x, from the definition of w and z, we have ux ≤ 0. So, from vt − ux = 0, we have that v(x, t) is decreasing with t for a.a. x ∈ R. s−1
2 Hence 0 ≤ v(x, t) ≤ v0 (x) ≤ ∥v0 ∥L∞ . We put λM = θ∥v0 ∥L∞ . Since wx , zx ≤ 0 and 0 ≤ v ≤ CM , by (2.2), we have
0 ≤ wt ≤ −λM wx and λM zx ≤ zt ≤ 0.
(2.4)
We set ρj,ε = ε−1 ρj (·/ε) as standard mollifiers with x and t for j = 1 and 2 respectively ∫∞ (ρj ∈ C0∞ (R) and ρj ≥ 0 and −∞ ρj (·)dx = 1 for j = 1, 2). We note that ∫
∞
0
ρ2,ε (t − s)ws (x, s)ds = ∂t
∫
∞
0
ρ2,ε (t − s)ws (x, s)ds + ρ2,ε (t)w(x, 0).
Hence, applying the mollifiers to the both side of the first inequality in (2.4), we have
where wε = we have
∫∞ 0
wε t − λM wε x + ρ2,ε (t)ρ1,ε ∗ w0 (x) ≤ 0, ρ2,ε (t − s)ρ1,ε ∗ w(x, s)ds. Noting wε (x + λM t, t) is differentiable with t, d wε (x + λM t, t) + ρ2,ε (t)ρ1,ε ∗ w0 (x + λM t) ≤ 0. dt
Integrating on [0, t], we have wε (x + λM t, t) − wε (x, 0) +
∫
t 0
ρ2,ε (s)ρ1,ε ∗ w0 (x + λM s)ds ≤ 0.
Since w(x, ·) and ∫w(·, t) are continuous∫with a.a. fixed t and x respectively, we have ∫∞ 0 t limε→0 wε (x, 0) → −∞ ρ2 (t)dtw0 (x) and 0 ρ2,ε (s)ρ1,ε ∗w(x+λM s, 0)ds → 0 ρ2 (t)dtw0 (x). Hence we have by taking ε → 0, w(x + λM t, t) − w0 (x) ≤ 0. Therefore we have w0 (x) ≤ w(x, t) ≤ w(x − λM t, 0) and z0 (x + λM t) ≤ z(x, t) ≤ z0 (x), which implies that (2.3). ∫∞ We put F (t) = − −∞ v(x, t) − v0 (x)dx. From the first equation (2.1) and Lemma 4, we have F (t) = (u− − u+ )t. We divide F (t) into the three parts as follows: F (t) = −
(∫
−M
−∞
+
∫
M
−M
+
∫
∞)
M
v(x, t) − v0 (x)dx =: F1 (t) + F2 (t) + F3 (t).
4
(2.5)
Now we estimate F1 (t). From the first equation in (2.1) and Lemma 4, we have ∫ −M d F1 (t) = −ux (x, t)dx = −u(t, −M ) + u− . dt −∞ Since w0 (−M ) ≤ w(−M, t), from the definition of w, we have −u(M, t) ≤ −v θ (−M, t) − w0 (−M ) ≤ −w0 (−M ) Hence we have F1 (t) ≤ −t(u− − w0 (−M )).
(2.6)
Since v ≥ 0 under our contradiction argument and v(t, x) ≤ v0 (x), we can estimate F2 (t) as F2 (t) ≤ 2M ∥v0 ∥L∞ .
(2.7)
In the same way as in the above estimate of F1 , we have F3 (t) ≤ t(−u+ + z0 (M )).
(2.8)
From (2.5), (2.6), (2.7) and (2.8), we have 2M ∥v0 ∥L∞ + (z0 (M ) − w0 (−M ) + u− − u+ )t ≥ (u− − u+ )t. Hence we have 2M ∥v0 ∥L∞ > (w0 (−M ) − z0 (M ))t,
which gives a contradiction for large t, since w0 (−M ) − z0 (M ) > 0. Furthermore, we have the estimate of the existence time of solutions. Therefore we complete the proof of Theorem 2. Remark 5. [Concluding remarks] • The proof of Theorem 1 is based on kinetic formulation by Lions, Perthame, Souganidis and Tadmor [4, 3] and the compensated compactness method. More precisely, we apply these theory to global solutions (wε,δ , zε,δ ) of the following approximated parabolic system: ε,δ ε,δ ε,δ wt + λ2 wx = εwxx s−1
ε,δ ztε,δ + λ1 zxε,δ = εzxx
s−1
with λ1 = −θ|v| 2 , λ2 = θ|v| 2 and initial data wε,δ (0, x) = w0 (x) − δ and z ε,δ (0, x) = z0 (x) + δ. If necessary, we apply the mollifier to initial data. For solutions (wε,δ , z ε,δ ), we can obtain necessary boundedness estimates. • Now we show that the solution constructed in Theorem 1 satisfies (1.11), (1.12) and 1.1 (R). We put (R, S) = (1.13), if w0 (x) and z0 (x) are bounded, decreasing and in Wloc (wxε,δ , zxε,δ ). Since (R, S) satisfies that Rt + (λ2 R)x = εRxx and St + (λ2 S)x = εSxx , we have from the negativity of R and S ∫ ∞ ∫ ∞ ∥wx (t)∥L1 + ∥zx (t)∥L1 = − R(x, t) + S(x, t)dx = − R(x, 0) + S(x, 0)dx −∞
= lim 2(u0 (−x) − u0 (x)). x→∞
5
−∞
The limx→∞ 2(u(−x) − u(x)) exists and is finite, since u0 = (w0 + z0 )/2 is decreasing and bounded. Hence, for solutions of the non-viscous equations (2.1), we have that 1,1 w(t, ·), z(t, ·) ∈ Wloc (R) for a.a. t ≥ 0 and that (1.12) is satisfied. Furthermore, in 1,1 the similar way as to the proof of Lemma 3, we have that w, z ∈ Wloc (R × R+ ) and (2.2) are satisfied. Therefore we have by the boundedness of λ1 and λ2 that ∥wt (t)∥L1 + ∥zt (t)∥L1 ≤ ∥λ2 wx (t)∥L1 + ∥λ2 zx (t)∥L1
≤ C(∥wx (0)∥L1 + ∥zx (0)∥L1 ).
Acknowledgments: This paper of the first author is partially supported by a Qianjiang professorship of Zhejiang Province of China, the National Natural Science Foundation of China (Grant No. 11271105). The second author is supported by Grant-in-Aid for Young Scientists Research (B), No. 16K17631.
References [1] N. Cristescu, Dynamic plasticity, North-Holland, Appl. Math. Mech., 1967. [2] F. James, Y.-J. Peng and B. Perthame, Kinetic formulation for chromatography and some other hyperbolic systems, J. Math. Pure Appl., 74 (1995), 367-385. [3] P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599-638. [4] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-system, Commun. Math. Phys., 163 (1994), 415-431. [5] Y.-G. Lu, Global H¨ older continuous solution of isentropic gas dynamics, Proc. Royal Soc. Edinburgh, 123A (1993), 231-238. [6] Y.-G. Lu, Hyperboilc Conservation Laws and the Compensated Compactness Method, Vol. 128, Chapman and Hall, CRC Press, New York, 2003. [7] Y.-G. Lu, Nonlinearly Degenerate Wave Equation, Rev. Acad. Colomb. Cienc.: Vol. XXXI, 119 (2007), 275-283. [8] Y. Sugiyama, Degeneracy in finite time of 1 D quasilinear wave equations, SIAM J. Math. Anal., 48 (2016), 847-860.
6
Yun-guang Lu, Yuusuke Sugiyama
PII: DOI: Reference:
S0893-9659(18)30143-5 https://doi.org/10.1016/j.aml.2018.05.001 AML 5508
To appear in:
Applied Mathematics Letters
Received date : 23 March 2018 Revised date : 1 May 2018 Accepted date : 1 May 2018 Please cite this article as: Y. Lu, Y. Sugiyama, Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity Yun-guang Lu∗and Yuusuke Sugiyama†
Abstract We prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations vtt = c(|v|s−1 v)xx with the nonnegative initial data v0 (x) and s > 1. These results are extensions of the results in the second author’s paper [Sugiyama SIAM J. Math. Anal., 48(2016), 847-860], where the existence and the∫ nonexistence of the unique global classical solution were studied ∞ with a threshold on −∞ v1 (x)dx and the non-degeneracy condition v0 (x) ≥ c0 > 0 on the initial data.
Key Words: Global weak solutions; degenerate wave equations; viscosity method; compensated compactness Mathematics Subject Classification 2010: 35L15, 35A01, 62H12.
1
Introduction
In this paper, we study the global generalized solutions of the nonlinearly degenerate wave equation vtt = c(|v|s−1 v)xx , −∞ < x < ∞, t > 0, (1.1) with the initial data (v, vt )|t=0 = (v0 (x), v1 (x)),
−∞ < x < ∞.
(1.2)
2
where v0 (x) ≥ 0, s > 1, c = θs > 0 and θ = s+1 2 > 0 are constants. The wave equation (1.1) ∫arises in the theory of elasticity and elasto-plasticity (see Cristescu [1]). v and x u = −∞ ∂t vdx describe the strain and the particle velocity and σ(v) = c|v|s−1 v s is the stress function. The exponent s in σ(v) is determined by materials. In our case s > 1, the stress function is applied to rubbers, soils and certain metals. In the case that 0 < s < 1, work hardening materials (see p. 15 in [1]). A function v ∈ L∞ (R × [0, T ]) is called a generalized solution of the Cauchy problem (1.1)-(1.2) if for any test function ϕ ∈ C0∞ (R × [0, ∞)), ∫ T∫ ∞ ∫ ∞ s−1 vϕ(x, t)tt − c|v| vϕ(x, t)xx dxdt + v0 (x)ϕ(x, 0)t − v1 (x)ϕ(x, 0)dx = 0. (1.3) 0
∗
−∞
−∞
K.K.Chen Institute for Advanced Studies Hangzhou Normal University, P. R. CHINA [email protected] † Corresponding author. Department of Mathematics, Tokyo University of Science Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan, e-mail:[email protected]
1
If we can take T = ∞ in (1.3), we call the function v ∈ L∞ (R × R+ ) global generalized solution of (1.3) Throughout this paper, we concentrate our study on the domain of v ≥ 0, then The Riemann invariants for (1.1) are given by w = u − vθ , z = u ˜ + vθ , where u=
∫
x
−∞
vt dx, u ˜=−
∫
∞
vt dx.
(1.4)
(1.5)
x
We denote w(x, 0) and z(x, 0) by w0 (x) and z0 (x) respectively. To state our main theorems, we define a threshold Γ as ∫ ∞ Γ= v1 (x)dx + v0θ (x)|x=+∞ + v0θ (x)|x=−∞ (1.6) −∞
We have the following first main result in this paper. Theorem 1. Let v0 (x) ≥ 0 be bounded, v1 (x) ∈ L1 (R) and the limits v0θ (x)|x=±∞ exist. Moreover suppose that Γ ≥ 0 and s−1
v1 (x) ± θv0 2 (x)∂x v0 (x) ≤ 0.
(1.7)
Then the Cauchy problem (1.1) and (1.2) has a global generalized solution satisfying (1.3). The proof of Theorem 1 is almost done in [7]. So we only give a comment on the proof of Theorem 1 in Remark 5. Our main contribution of this paper is the second main theorem, which implies the nonexistence of generalized solutions satisfying v ≥ 0. Furthermore, Theorem 2 give a upper estimate of the maximal existence time of solutions, which is not given in the proof of [8]. In order to state the second main theorem, we collect some conditions for global solutions. We can easily check that the global generalized solutions of (1.1) constructed Theorem 1 satisfy that v(x, t) ≥ 0 for a.a. (x, t) ∈ R × R+
(1.8)
w(x, t) and z(x, t) are decreasing with x for a.a. t ≥ 0,
(1.9)
w, z ∈ L∞ (R × R+ ).
(1.10)
Furthermore, we can show that the following properties are also satisfied for the global solutions in Theorem 1, if we assume the additional regularity on initial data that w0 , z0 ∈ 1,1 Wloc (R): 1,1 w, z ∈ Wloc (R × R+ ) and ∂t v(t, ·) ∈ L1 (R),
(1.11)
∥wx (t)∥L1 + ∥zx (t)∥L1 ≤ ∥wx (0)∥L1 + ∥zx (0)∥L1 ,
(1.12)
∥wt (t)∥L1 + ∥zt (t)∥L1 ≤ C(∥wx (0)∥L1 + ∥zx (0)∥L1 )
(1.13)
and
for a.a. t ≥ 0. We have the following second main result in this paper. 2
1,1 (R) ∩ L∞ (R) and v1 ∈ L1 (R). Suppose that Theorem 2. Let v0 ≥ 0, w0 , z0 ∈ Wloc the limits v0θ (x)|x=±∞ exist. Furthermore, we assume that (1.7) is satisfied and Γ < 0. Then the Cauchy problem (1.1) and (1.2) has no solutions satisfying the properties (1.8)(1.13). Furthermore, solutions satisfying (1.8)-(1.13) can not exist beyond the time T ∗ = ∫M 2M ∥u0 ∥L∞ /−Γ where M is a positive constant such that −M v1 (x)dx+v0θ (M )+v0θ (−M ) < 0.
Theorems 1 and 2 extend the results in [8]. In [8], under the assumption (1.7), the second author has already obtained the threshold Γ separating the global existence of solutions and the occurrence of the degeneracy in finite time. However in [8], it is assume that initial data are sufficiently smooth and that v0 (x) ≥∃ c0 > 0 for all x ∈ R. We note that the hyperbolicity is lost when the equation degenerate. Under the assumption that solutions are nonnegative, the decreasing property of w0 and z0 ensures the absence of shock waves. So we can expect that generalized solutions do not satisfy the regularity properties (1.11)-(1.13) are not satisfied, if nonnegativity of solutions does not hold. This expectation is a motivation for Theorem 2. The main theorem in [8] can not estimate the time when the degeneracy occurs. Furthermore, non-smooth solutions are not treated in [8], since the proof of the main theorem is based on the method of characteristics. While H¨older continuous, BV or L∞ solutions has been discussed in many papers (e.g. [5, 6, 7] and Lions, Perthame, Souganidis and Tadmor [4, 3]) ∫ ∞for 1D hyperbolic conservation law. The key for the proof in [8] is the function F (t) = −∞ v(x, 0) − v(x, t)dx In the estimate for F (t), we divide the integral region (−∞, ∞) of F by three parts, using characteristic curves. However, for non-classical solutions, the characteristic curves would not be defined. Observing 0 ≤ wt (x, t) and zt (x, t) ≤ 0, we estimate F more simply.
2
Proof of Theorem 2
∫M 1,1 We note that w0 , z0 ∈ C(R) ⊂ Wloc (R) and −M v1 (x)dx + v0θ (M ) + v0θ (−M ) = w0 (M ) − z0 (−M ). Hence, since w0 and z0 are decreasing, the assumption Γ < 0 implies that lim w0 (x) ≥ w0 (x) > z0 (y) ≥ lim z( x). x→∞
x→−∞
Hence there exists a large constant M0 > 0 such that if M ≥ M0 , then w0 (−M ) > z0 (M ). Lemma 3. Suppose that the assumptions (1.8)-(1.13) are satisfied. Then we have for a.a. (x, t) ∈ R × R+ . { vt − ux = 0, (2.1) ut − c(|v|s−1 v)x = 0 and
{
wt + λ2 wx = 0, zt + λ1 zx = 0.
(2.2)
1,1 This Lemma states that for wloc solutions, the weak form of the equation can be reduced to a 2 × 2 hyperbolic system (not weak sense) under the assumptions (1.8)-(1.13). The proof is based on standard density argument. So we omit it. To state the next lemma, we put limx→±∞ v0 (x) = v± .
3
Lemma 4. Suppose that the assumptions (1.8)-(1.13) is satisfied, then we have lim v(x, t) = v± .
(2.3)
x→±∞
Proof. Since w and z are decreasing with x, from the definition of w and z, we have ux ≤ 0. So, from vt − ux = 0, we have that v(x, t) is decreasing with t for a.a. x ∈ R. s−1
2 Hence 0 ≤ v(x, t) ≤ v0 (x) ≤ ∥v0 ∥L∞ . We put λM = θ∥v0 ∥L∞ . Since wx , zx ≤ 0 and 0 ≤ v ≤ CM , by (2.2), we have
0 ≤ wt ≤ −λM wx and λM zx ≤ zt ≤ 0.
(2.4)
We set ρj,ε = ε−1 ρj (·/ε) as standard mollifiers with x and t for j = 1 and 2 respectively ∫∞ (ρj ∈ C0∞ (R) and ρj ≥ 0 and −∞ ρj (·)dx = 1 for j = 1, 2). We note that ∫
∞
0
ρ2,ε (t − s)ws (x, s)ds = ∂t
∫
∞
0
ρ2,ε (t − s)ws (x, s)ds + ρ2,ε (t)w(x, 0).
Hence, applying the mollifiers to the both side of the first inequality in (2.4), we have
where wε = we have
∫∞ 0
wε t − λM wε x + ρ2,ε (t)ρ1,ε ∗ w0 (x) ≤ 0, ρ2,ε (t − s)ρ1,ε ∗ w(x, s)ds. Noting wε (x + λM t, t) is differentiable with t, d wε (x + λM t, t) + ρ2,ε (t)ρ1,ε ∗ w0 (x + λM t) ≤ 0. dt
Integrating on [0, t], we have wε (x + λM t, t) − wε (x, 0) +
∫
t 0
ρ2,ε (s)ρ1,ε ∗ w0 (x + λM s)ds ≤ 0.
Since w(x, ·) and ∫w(·, t) are continuous∫with a.a. fixed t and x respectively, we have ∫∞ 0 t limε→0 wε (x, 0) → −∞ ρ2 (t)dtw0 (x) and 0 ρ2,ε (s)ρ1,ε ∗w(x+λM s, 0)ds → 0 ρ2 (t)dtw0 (x). Hence we have by taking ε → 0, w(x + λM t, t) − w0 (x) ≤ 0. Therefore we have w0 (x) ≤ w(x, t) ≤ w(x − λM t, 0) and z0 (x + λM t) ≤ z(x, t) ≤ z0 (x), which implies that (2.3). ∫∞ We put F (t) = − −∞ v(x, t) − v0 (x)dx. From the first equation (2.1) and Lemma 4, we have F (t) = (u− − u+ )t. We divide F (t) into the three parts as follows: F (t) = −
(∫
−M
−∞
+
∫
M
−M
+
∫
∞)
M
v(x, t) − v0 (x)dx =: F1 (t) + F2 (t) + F3 (t).
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(2.5)
Now we estimate F1 (t). From the first equation in (2.1) and Lemma 4, we have ∫ −M d F1 (t) = −ux (x, t)dx = −u(t, −M ) + u− . dt −∞ Since w0 (−M ) ≤ w(−M, t), from the definition of w, we have −u(M, t) ≤ −v θ (−M, t) − w0 (−M ) ≤ −w0 (−M ) Hence we have F1 (t) ≤ −t(u− − w0 (−M )).
(2.6)
Since v ≥ 0 under our contradiction argument and v(t, x) ≤ v0 (x), we can estimate F2 (t) as F2 (t) ≤ 2M ∥v0 ∥L∞ .
(2.7)
In the same way as in the above estimate of F1 , we have F3 (t) ≤ t(−u+ + z0 (M )).
(2.8)
From (2.5), (2.6), (2.7) and (2.8), we have 2M ∥v0 ∥L∞ + (z0 (M ) − w0 (−M ) + u− − u+ )t ≥ (u− − u+ )t. Hence we have 2M ∥v0 ∥L∞ > (w0 (−M ) − z0 (M ))t,
which gives a contradiction for large t, since w0 (−M ) − z0 (M ) > 0. Furthermore, we have the estimate of the existence time of solutions. Therefore we complete the proof of Theorem 2. Remark 5. [Concluding remarks] • The proof of Theorem 1 is based on kinetic formulation by Lions, Perthame, Souganidis and Tadmor [4, 3] and the compensated compactness method. More precisely, we apply these theory to global solutions (wε,δ , zε,δ ) of the following approximated parabolic system: ε,δ ε,δ ε,δ wt + λ2 wx = εwxx s−1
ε,δ ztε,δ + λ1 zxε,δ = εzxx
s−1
with λ1 = −θ|v| 2 , λ2 = θ|v| 2 and initial data wε,δ (0, x) = w0 (x) − δ and z ε,δ (0, x) = z0 (x) + δ. If necessary, we apply the mollifier to initial data. For solutions (wε,δ , z ε,δ ), we can obtain necessary boundedness estimates. • Now we show that the solution constructed in Theorem 1 satisfies (1.11), (1.12) and 1.1 (R). We put (R, S) = (1.13), if w0 (x) and z0 (x) are bounded, decreasing and in Wloc (wxε,δ , zxε,δ ). Since (R, S) satisfies that Rt + (λ2 R)x = εRxx and St + (λ2 S)x = εSxx , we have from the negativity of R and S ∫ ∞ ∫ ∞ ∥wx (t)∥L1 + ∥zx (t)∥L1 = − R(x, t) + S(x, t)dx = − R(x, 0) + S(x, 0)dx −∞
= lim 2(u0 (−x) − u0 (x)). x→∞
5
−∞
The limx→∞ 2(u(−x) − u(x)) exists and is finite, since u0 = (w0 + z0 )/2 is decreasing and bounded. Hence, for solutions of the non-viscous equations (2.1), we have that 1,1 w(t, ·), z(t, ·) ∈ Wloc (R) for a.a. t ≥ 0 and that (1.12) is satisfied. Furthermore, in 1,1 the similar way as to the proof of Lemma 3, we have that w, z ∈ Wloc (R × R+ ) and (2.2) are satisfied. Therefore we have by the boundedness of λ1 and λ2 that ∥wt (t)∥L1 + ∥zt (t)∥L1 ≤ ∥λ2 wx (t)∥L1 + ∥λ2 zx (t)∥L1
≤ C(∥wx (0)∥L1 + ∥zx (0)∥L1 ).
Acknowledgments: This paper of the first author is partially supported by a Qianjiang professorship of Zhejiang Province of China, the National Natural Science Foundation of China (Grant No. 11271105). The second author is supported by Grant-in-Aid for Young Scientists Research (B), No. 16K17631.
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