ON AN INEQUALITY IN NONLINEAR
THERMOELASTICITY
Inequality in the Nonlinear Elasticity
VLADIMIR JOVANOVIĆ
Faculty of Sciences
Mladena Stojanovića 2
78000 Banja Luka
38039 Bosnia and Herzegovina
EMail: vladimir@mathematik.uni-freiburg.de
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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16 November, 2007
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Communicated by:
S.S. Dragomir
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2000 AMS Sub. Class.:
26D15, 35L45, 74B20.
Key words:
Integral inequality, Elastodynamics, Lax – Friedrichs scheme.
Abstract:
This paper deals with an integral inequality which arises in numerical analysis of
the Lax – Friedrichs scheme for the elastodynamics system. It is obtained as a
consequence of a more general inequality.
Received:
15 November, 2006
Accepted:
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Contents
1
Introduction
3
2
Lax – Friedrichs Scheme for the Elastodynamics System
5
3
Proof of the Inequalities
7
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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1.
Introduction
Let us consider the following problem:
Theorem 1.1. Let a, b ∈ R, a < 0, b > 0 and f ∈ C[a, b], such that:
(1.1)
0 < f ≤ 1 on [a, b],
(1.2)
f is decreasing on [a, 0],
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
Z
0
b
Z
f dx =
(1.3)
a
f dx.
0
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then
Z
b
2
Z
f dx ≤ 2
(1.4)
a
a+b
2
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f dx.
a
As we will see later, Theorem 1.1 is a transformed and slightly generalized form
of a problem related to the numerical analysis of a nonlinear system of PDEs. This
problem is stated below.
Theorem 1.2. Suppose that σ ∈ C 2 (R) satisfies
(1.5)
(1.6)
vol. 8, iss. 4, art. 105, 2007
σ 0 (w) > 0 for all w ∈ R
w σ 00 (w) > 0 for all w ∈ R\{0}.
Assume further that for w1 , w2 ∈ [−1, ∞), w1 < 0, w2 > 0 and α > 0, the conditions
Z w2 √
Z 0√
0
σ ds =
σ 0 ds,
(1.7)
w1
0
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and
(1.8)
α
p
σ 0 (w) ≤ 1
for all w ∈ [w1 , w2 ].
hold. Then
Z
w1 +w2
2
(1.9)
w1
√
σ 0 ds ≥
α
σ(w2 ) − σ(w1 ) .
2
The main subject of this paper is the inequality (1.9). In the next section we
describe the context in which the inequality arises. We start the third section with
the proof of Theorem 1.1, then proceed with the proof of Theorem 1.2 and finally
conclude the section with two remarks.
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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2.
Lax – Friedrichs Scheme for the Elastodynamics System
The elastodynamics system governs isentropic processes in thermoelastic nonconductors of heat. The Cauchy problem for the underlying system in the one-dimensional
case has the form
(2.1)
∂t w − ∂x v = 0,
∂t v − ∂x σ(w) = 0 in R × (0, T ),
Inequality in the Nonlinear Elasticity
(2.2)
w(x, 0) = w0 (x),
Vladimir Jovanović
v(x, 0) = v0 (x) in R,
vol. 8, iss. 4, art. 105, 2007
where w : R × [0, T ) → [−1, ∞) is the strain and v : R × [0, T ) → R is the
velocity. In the theory of nonlinear systems of conservation laws this system plays
an important role due to its accessability to a detailed mathematical analysis (see [1]).
The special feature that renders these equations amenable to analytical treatment is
the existence of the so-called compact invariant regions. Invariant regions are sets
S ⊂ R2 with the following property: if the initial function u0 = (w0 , v0 ) takes its
values in S, then so does the solution u = (w, v) of (2.1), (2.2). It can be shown
(see [1]) that for N > 0, the sets given by
(2.3)
SN = {(w, v) ⊂ [−1, ∞) × R : |y(w, v)| ≤ N, |z(w, v)| ≤ N },
are invariant for the Cauchy problem (2.1), (2.2), where
Z wp
Z
0
y(w, v) = −
σ (s) ds + v,
z(w, v) = −
w0
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w
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p
σ 0 (s) ds − v
w0
are the the so-called Riemann invariants.
The Lax – Friedrichs scheme is frequently used as a discretization procedure for
systems of conservation laws. In our particular case, the scheme takes the form
1
α
(2.4)
un+1
= uni −
f (uni+1 ) − f (uni−1 ) + uni−1 − 2uni + uni+1 ,
i
2
2
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where α > 0 is a parameter and uni = (win , vin ) for i ∈ Z, n ∈ N. Here we
used f (u) = (−v, −σ(w)), with u = (w, v). For the numerical stability of the
Lax – Friedrichs scheme it is crucial that the sets SN from (2.3) are also invariant
for (2.4). That is, if unip∈ SN for all i ∈ Z, then un+1
∈ SN for all i ∈ Z,
i
0
provided α · sup(w,v)∈SN σ (w) ≤ 1, (see [3]). Similarly as in [2], the proof of the
invariancy is reduced to some problems associated with certain integral inequalities.
The problem stated in Theorem 1.2 is one of them.
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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3.
Proof of the Inequalities
Proof of Theorem 1.1. We will consider two cases.
1. Case: a + b ≥ 0.
By (1.1) and (1.3), we have
Z b
Z b
Z
2
f dx ≤
f dx = 2
a
a
0
Z
f dx ≤ 2
a+b
2
f dx.
a
a
2. Case: a + b < 0.
First, note that due to (1.1) and (1.3), for every a0 ∈ [a, 0] there exists a unique
R0
R b0
b0 ∈ [0, b], such that a0 f dx = 0 f dx. Therefore, one can introduce a function
R0
R ϕ(x)
ϕ : [a, 0] → [0, b] with the property x f ds = 0 f dx. Obviously, ϕ(a) = b
and ϕ(0) = 0. It is a simple matter to prove that ϕ is differentiable and that for all
x ∈ [a, 0],
f (ϕ(x)) ϕ0 (x) = −f (x).
(3.1)
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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We will show that the inequality
Z
(3.2)
x
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ϕ(x)
f 2 ds ≤ 2
Z
x+ϕ(x)
2
f ds,
x
holds for all x ∈ [a, 0]. Then (1.4) will be a consequence of (3.2), when x = a.
Let x ∈ [a, 0] be arbitrary. If x+ϕ(x) ≥ 0, then we proceed as in Case 1. Therefore,
suppose that
(3.3)
x + ϕ(x) < 0.
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Define a function ψ : [a, 0] → R with
x+ϕ(x)
2
Z
ψ(x) = 2
Z
ϕ(x)
f ds −
x
f 2 ds.
x
From
0
0
ψ (x) = (1 + ϕ (x)) f
x + ϕ(x)
2
− 2f (x) − f 2 (ϕ(x)) ϕ02 (x),
Inequality in the Nonlinear Elasticity
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
using (3.1) follows
x + ϕ(x)
2
− 2f (x) f (ϕ(x)) + f 2 (ϕ(x)) f (x) + f 2 (x) f (ϕ(x)).
0
f (ϕ(x)) ψ (x) = f (ϕ(x)) − f (x) f
If f (ϕ(x))−f (x) ≤ 0, then obviously ψ 0 (x) ≤ 0. Assume now f (ϕ(x))−f (x) > 0.
x+ϕ(x)
Using the fact that x ≤ 0, ϕ(x)
≥ 0 and (3.3), we obtain 0 > 2 ≥ x, which
together with (1.2) yields, f
x+ϕ(x)
2
≤ f (x). Hence,
f (ϕ(x)) ψ 0 (x) ≤ f (ϕ(x)) − f (x) f (x) − 2f (x) f (ϕ(x))
+ f 2 (ϕ(x)) f (x) + f 2 (x) f (ϕ(x))
= f (x) f (x) + f (ϕ(x)) f (ϕ(x)) − 1 ≤ 0.
Hence we have shown that ψ 0 (x) ≤ 0 for all x ∈ [a, 0]. Since ψ(0) = 0, one
concludes that ψ ≥ 0 on [a, 0], that is, (3.2) holds.
Rw
Proof of Theorem 1.2. Since σ(w2 ) − σ(w1 ) = w12 σ 0 ds, then by multiplying (1.9)
√
by 2α and introducing f = α σ 0 , the inequality (1.9) is transformed into (1.4), with
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a = w1 , b = w2 . Due to (1.6), σ 0 decreases on [w1 , 0], so f does on [a, 0], as well.
The relations (1.5) and (1.8) yield (1.1). The equality (1.7) implies (1.3). Therefore,
Theorem 1.1 applies.
Remark 1. Assume that (1.1), (1.2) and (1.3) hold for f ∈ C[a, b].
(a) The constant A = 2 in
Z
b
2
f dx ≤ A
(3.4)
a
Inequality in the Nonlinear Elasticity
a+b
2
Z
f dx
a
is optimal in the case a + b = 0; indeed, taking f = 1 in (3.4), one obtains
A ≥ 2.
(b) It is easy to see that if p ≥ 2, then the inequality
Z b
Z a+b
2
p
(3.5)
f dx ≤ Ap
f dx
a
a
holds for all Ap ≥ 2. However, if 1 ≤ p < 2, then proceeding similarly as in
the proof of Theorem 1.1, one can deduce that (3.5) is satisfied for all Ap ≥ 4.
Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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References
[1] C. DAFERMOS, Hyperbolic Conservation Laws in Continuum Physics, Berlin
Heidelberg New York: Springer–Verlag (2000)
[2] D. HOFF, A finite difference scheme for a system of two conservation laws with
artificial viscosity, Math. Comput., 33(148) (1979), 1171–1193.
[3] V. JOVANOVIĆ AND C. ROHDE, Error estimates for finite volume approximations of classical solutions for nonlinear systems of balance laws, SIAM J. Numer. Anal., 43 (2006), 2423–2449.
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Vladimir Jovanović
vol. 8, iss. 4, art. 105, 2007
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