Nonlinear
L
2-stability under large disturbances
1 Paulo R. Zingano∗Instituto de Matematica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil Received 1 September 1997; received in revised form 11 March 1998
Abstract
We derive time-asymptotic decay rates inL2 for large disturbances to some important classes of solutions of the Cauchy problem for a number of uniformly parabolic equations, provided only that the disturbances belong to appropriateLpspaces at initial time. Examples considered include the scalar nonlinear advection–diusion equation
ut+f(u)x= (b(u)ux)x and the parabolic system
ut+ (u’(|u|))x= (B(u)ux)x;
whereu(x; t)∈Rm; ’is a given scalar function andB(u) is a uniformly positive-denite diagonal matrix. c1999 Elsevier Science B.V. All rights reserved.
Keywords:Nonlinear stability; Decay rates; L2 estimates; Uniformly parabolic equations; Cauchy problem; Diusion waves; Rarefaction waves
1. Introduction
One of the basic questions concerning systems of conservation laws is the large time behavior of disturbances to certain fundamental classes of solutions, like shocks or traveling waves, expansion waves and equilibrium or constant solutions. Under appropriate assumptions, these waves are non-linearly stable, and disturbances decay time asymptotically provided that they are suciently small (in an appropriate sense) at initial time. For example, considering the parabolic system
ut+f(u)x= (B(u)ux)x;
where u(·; t) deviates at t= 0 from a given constant solution, which, without loss of generality, we assume to be the zero state, a detailed description of the asymptotic behavior of u(·; t) as t→ ∞
1This work was partly supported by CNPq and FAPERGS, Brazil.
∗E-mail: zingano@mat.ufrgs.br.
has been given when the Jacobian matrix f′
(u) is completely hyperbolic [3, 4, 13], provided u(·;0) is small enough to satisfy
Z +∞
−∞
|u(x;0)|(1 +|x|) dx+ Z +∞
−∞
(|u(x;0)|2+|u
x(x;0)|2) dx6;
for≪1. Here,u(x; t) denotes anm-dimensional vector of unknown quantities (u1(x; t); : : : ; um(x; t));
f(u(x; t)) is a vector function giving the ux of the conserved variables at the location x and time
t, and B(u) is a positive-denite viscosity matrix describing the viscous dissipation present in the system. In particular, we get from their results the estimate
ku(·; t)kL2(R)= O(t−1=4); (1)
where the decay rate t−1=4 is optimal, since solutions can be found which decay exactly at this rate
[4]. Using much weaker assumptions on the initial data, we derive this estimate here for a class of systems which includes
ut+ (u’(|u|))x= (B(u)ux)x; (2)
where ’ is a scalar function and B(u) is a positive-denite diagonal matrix for all values of u
concerned. This is possible due to the particular structure of the solutions of (2). One example is the rotationally invariant system [7, 18]
ut+ (u|u|2)x=uxx; (3)
where is a positive constant. The inviscid form of Eq. (3) was considered in 1979 by Keytz and Kranzer in connection with the elastic string problem in elasticity [14]. This system has also been studied in one-dimensional multiphase ow [12, 17], magnetohydrodynamics [1, 6] and more generally in continuum mechanics as a basic model for the propagation of plane waves in isotropic, multidimensional systems [1, 2]. For these and more general systems like (2), we show in Section 2 that
ku(·; t)kL2(R)6C(1 +t)−1=4; (4)
wheneveru(·;0)∈L1(
R)∩L2(
R), however large, where C is a positive constant which depends only on the magnitude of ku(·;0)kL1(R) andku(·;0)kL2(R), the dimension parameter m and ¿0 such that
h; B(u)i¿||2 ∀∈
Rm (5)
for alluconcerned. Thus, the solutionu=0 of Eq. (2) is asymptotically stable under arbitrarily large disturbances in L1(
R)∩L2(
R). A similar property is shown to hold for the scalar advection–diusion equation
ut+f(u)x= (b(u)ux)x; (6)
where b(u) is positive. Given an arbitrary initial state u(·;0)∈L1(R)∩L∞(
R), it is shown in Section 3 that, for any t ¿0,
for some constant C which depends only on the magnitude of ku(·;0)kL1(R) and ku(·;0)kL∞
(R), and
¿0 such that
b(!)¿ for |!|6ku(·;0)kL∞(
R): (8)
Finally, in Section 4, we simplify the analysis in [19] to establish a similar result for the time decay in L2-norm of arbitrarily large disturbances to scalar rarefaction waves in case of a quadratic ux,
i.e., Burgers equation [5, 10]. This time we are concerned with the large time behavior of solutions of the initial value problem
ut+f(u)x=uxx; (9a)
u(x;0) =u0(x); (9b)
where
f(u) =au2+bu+c (10)
and the initial prole u0 is a bounded measurable function connecting two given constant states v±
at ±∞, in the sense that
Z 0
−∞
|u0(x)−v−|dx ¡∞ (11a)
and
Z +∞
0
|u0(x)−v+|dx ¡∞: (11b)
We denote byB(v−; v+) the space of all such functions, i.e., allu0∈L∞(R) for which both integrals
in (11a) and (11b) are nite. In (10), we assume for deniteness that we have a ¿0, so that our interest here is the case when
v−¡ v+: (12)
Taking an initial state v(·;0)∈B(v−; v+) which is monotonically increasing, the corresponding
so-lution v(x; t) of problem (9) will stay monotonic in x for all t ¿0, see, e.g., [19], and is called a (viscous) rarefaction or expansion wave. Hence, associated with (9)–(12) above, we consider the problem
vt+f(v)x=vxx; (13a)
v(x;0) =v0(x); (13b)
where v0∈B(v−; v+) is monotonic. Thus, we may regard the initial state u(·;0) as a perturbation of
v(·;0),
u(x;0) =v(x;0) +(x); (14)
where the disturbance satises
∈L1(
R)∩L∞
but is otherwise arbitrary. Under the conditions above, the solution u(·; t) is shown to converge to the rarefaction wave v(·; t) as t→ ∞, with
ku(·; t)−v(·; t)kL2(R)6C(1 +t)
−1=4; (16)
where the constant C depends only on the magnitude of kkL1(R) andkkL∞
(R) and a ¿0 is given in
(10). Thus, like the previous examples, the rarefaction waves (13) exhibit asymptotic stability in L2
with respect to arbitrarily large disturbances over their initial prole, provided that they are bounded and integrable. This is also true for other Lp-norms, p ¿1, as can be shown using standard energy methods to estimate kux(·; t)−v(·; t)kL2(R) once the results of Section 4 have been proved. We note
that the L∞
-stability was rst shown in [11], without a decay rate. Indeed, they showed that
ku(·; t)−r(·; t)kL∞
(R)→0 as t→ ∞; (17)
where r(x; t) is the self-similar, entropy solution of the inviscid problem
rt+f(r)x= 0; (18a)
r(x;0) = v
−; x ¡0;
v+; x ¿0:
(18b)
When f is quadratic, a very detailed description of the viscous waves u(·; t) has been given [9], which yields
ku(·; t)−r(·; t)kLp(R)= O(1)t
−1=2+1=2p: (19)
This can be generalized to more general ux functions using the methods given here and in [8] to estimate ku(·; t)−v(·; t)kLp(R) and kv(·; t)−r(·; t)kLp(R), respectively, yielding in the case of convex
ux the estimate
ku(·; t)−r(·; t)kLp(R)= O(1)(logt)1=2+1=2pt−1=2+1=2p:
In what follows, we will derive the estimates (4), (7) and (16) adapting the discussion in [18– 20] to our present needs. In particular, we will extend the argument in [18] to the more general systems (2), and make use of (10) to give a direct derivation of (16) which is much simpler than the general treatment considered in [19]. (One should note, however, that for a general convex ux f the derivation given here would require the additional condition that the disturbance be suciently small in L1(R).) This approach avoids many technicalities which are dealt with in the
references above and brings out more clearly the similarities in the analysis for each case. As to the notation used in the text, boldface characters will always denote vector quantities, while capital letters will be usually reserved for matrices. A symbol like CA will denote a constant whose value
depends on a set of parameters specied by the list A; distinct references to the same constant
symbol will not necessarily imply the same numerical value, so that we will write 2CA again
2. A class of parabolic systems
We will consider in this section the L2 decay of disturbances to the equilibrium solution u=0 of
the parabolic system
ut+ (u’(|u|))x= (B(u)ux)x; (20)
where ’ denotes a continuously dierentiable scalar function, x∈R; t ¿0; u(x; t) is the vector of unknowns (u1(x; t); : : : ; um(x; t)), and B(u) is an m×m diagonal matrix which is uniformly positive denite, i.e.,
B(u) = diag{bi(u); i= 1; : : : ; m}; (21a)
bi(u)¿ ∀i= 1; : : : ; m (21b)
for all u∈Rm; where is a positive constant. The initial state u(·;0) is any Lebesgue measurable pulse with nite mass and energy, i.e., u(·;0)∈L1(R)∩L2(R), and we let K ¿0 be suciently
large so that
ku(·;0)kL1(R)+ku(·;0)kL2(R)6K: (22)
Under these conditions, we will show in this section that there exists a positive constant CK,
depending only on the parameters K={m; K; }, such that
ku(·; t)kL2(R)6CK(1 +t)
−1=4 (23)
for all t ¿0. We will prove this estimate in the following way. First, we note that the solution operator of (20) is L1-contractive, i.e., we have
ku(·; t)kL1(R)6ku(·;0)kL1(R) (24)
for any t ¿0, where
ku(·; t)kL1(R)=ku1(·; t)kL1(R)+· · ·+kum(·; t)kL1(R)
and similarly for ku(·;0)kL1(R). In fact, more is true: Eq. (20) is L1-contractive in each component
ui(·; t) individually, i.e., kui(·; t)kL1(R)6kui(·;0)kL1(R) for all t ¿0 and i= 1;2; : : : ; m. This can be
proved in a standard way as in [8, 15], but for convenience of the reader we will briey review the argument. Taking a regularized sign function L′
(see, e.g., [15, 19]), we multiply the ith component of Eq. (20) by L′
(ui(x; t)) and integrate the result over R×[0; T] to get, after a few integrations by parts,
Z +∞
−∞
L(ui(x; T)) dx+ Z T
0
Z +∞
−∞
L′′(ui(x; t))bi(u(x; t)) @u
i
@x
2 dxdt
= Z +∞
−∞
L(ui(x;0)) dx+ Z T
0
Z +∞
−∞
L′′(ui(x; t))ui(x; t)@ui
Since bi(u) is nonnegative, we then have
by Lebesgue’s dominated convergence theorem, which concludes the derivation of inequality (24) above. Another property which can be easily derived is the following energy estimate:
ku(·; T)k2 summing from i= 1 to m, we get, after a few similar computations,
m
from which we immediately get (25), since
=
Using the elementary Sobolev inequality
kui(·; t)kL2(R)6Ckui(·; t)k2=3
for each t ¿0, whereCm; K denotes a constant which depends only on m; K. Hence, we obtain from (27) the estimate
Since, by Holder’s inequality, we have
Z T
we get from (28) that
Y(T)6CK{1 + (1 +T)1=3Y(T)1=3}
for some constant CK which depends on K={m; K; } given in (21) and (22). This immediately
gives
for some suitable constant CK which, again, depends on K={m; K; }. Recalling the denition of Y(T) given in (29) above, we then have
(1 +T)ku(·; T)k2
L2(R)+
Z T
0
(1 +t)kDu(·; t)k2
L2(R)dt6CK(1 +T)1=2:
A little more generally, given ¿12, if we repeat the derivation above using (1 +t) instead of (1 +t), we will arrive at the following result.
Theorem 1. Let u(x; t) be the solution of Eq. (20) corresponding to an initial prole u(·;0) in L1(R)∩L2(R). Then; given any ¿1
2; there exists a constant C;K (depending only on and
K={m; K; } given in (21) and (22)) such that
(1 +T)ku(·; T)k2
L2(R)+
Z T
0
(1 +t)kDu(·; t)k2
L2(R)dt6C;K(1 +T)
−1=2
for every T ¿0.
3. The scalar advection–diusion equation
In this section we will establish by a similar argument the L2 decay of solutions u(x; t) of the
equation
ut+f(u)x= (b(u)ux)x; (31)
where u(·;0) is an arbitrary bounded, integrable pulse on the real line, i.e.,
u(·;0)∈L1(R)∩L∞(R): (32)
Similarly to (21) above, we assume in this section that
b(u)¿ (33)
for all u concerned, where is a positive constant. Under these assumptions, we will derive the estimate
ku(·; t)kL2(R)6C(1 +t)−1=4 (34)
for all t ¿0, where C is a constant which depends on and the magnitude of ku(·;0)kL1(R) and
ku(·;0)kL∞
(R), referring the reader to [20] for the corresponding analysis in the n-dimensional case.
Given u(·;0) in L1(R)∩L∞(
R), we let K ¿0 be large enough such that
ku(·;0)kL1(R)+ku(·;0)kL∞
(R)6K: (35)
It is well known that, for any t ¿0, u(x; t) satises the maximum principle [16]
ku(·; t)kL∞(
R)6ku(·;0)kL∞(
R) (36)
and the L1-contractive property
which can be established in a way similar to (24) above. In order to show (34), given T ¿0
so that, using (33), we obtain
(1 +T)ku(·; T)k2
From (35) and (37) and the Sobolev inequality
ku(·; t)kL2(R)6Cku(·; t)k2=3
where CK is a constant which depends on K. Hence, (38) yields
(1 +T)ku(·; T)k2
the previous section, we use Holder’s inequality to get
Z T
so that, from (39), we obtain
(1 +T)ku(·; T)k2
Theorem 2. Given ¿1
2; the solution u(x; t) of (31)–(33) satises; for all T ¿0;
(1 +T)ku(·; T)k2
L2(R)+
Z T
0
(1 +t)kDu(·; t)k2
L2(R)dt;
6C;K(1 +T)
−1=2
for some constant C;K which depends only on and K={K; } given in (33) and (35).
4. Rarefaction waves
Let f be the quadratic function given in (10), where we assume a ¿0, and that u(x; t) is the solution of the initial value problem
ut+f(u)x=uxx; (40a)
u(x;0) =v0(x) +(x); (40b)
where v0 is a monotonically increasing prole connecting two given constant states v−¡ v+ as
x→ ± ∞; so that (11) is satised, i.e.,
Z 0
−∞
|v0(x)−v−|dx ¡∞; (41a)
and
Z +∞
0
|v0(x)−v+|dx ¡∞: (41b)
The disturbance is assumed to be bounded and integrable, ∈L1(R)∩L∞(
R), but is otherwise arbitrary. As before, we take K ¿0 suciently large so that
kkL1(R)+kkL∞
(R)6K: (42)
Associated with (40) above, we consider the initial value problem
vt+f(v)x=vxx; (43a)
v(x;0) =v0(x): (43b)
Becausev(·;0)∈B(v−; v+) is monotonic, it is well known (see, e.g., [19]) thatv(·; t) stays monotonic
for all t ¿0, and is called a (viscous) rarefaction or expansion wave. Moreover, v(·; t) satises
kvx(·; t)kL∞ (R)6
1
at ∀t ¿0; (44)
where a=f′′
(v), see (10), so that the rarefaction wave attens out at a linear rate as t increases [19]. Another important property is given by
since the solution operator of (40) is L1-contractive [8, 19]. Under the conditions above, it will be
shown in this section that the solution u(·; t) converges in L2(R) to the rarefaction wave v(·; t) as
t→ ∞, with
ku(·; t)−v(·; t)kL2(R)6Ca; K(1 +t)−1=4 (46)
for all t ¿0, whereCa; K denotes a constant which depends on aand K. Once (46) has been proved, it would be easy to derive by standard energy methods the more general estimate
ku(·; t)−v(·; t)kLp(R)6Ca; K(1 +t)−1=2+1=2p
for each 16p6∞ and t ¿0, where Ca; K depends on a, K but not on p. Hence, the rarefaction wave (43) is asymptotically stable in Lp(
R), p ¿1, with respect to arbitrarily large disturbances , provided only that belongs to L1(R)∩L∞(
R). For a generalization of this result to an arbitrary convex ux f, we refer the reader to [19]. In the sequel, we will establish (46) following an argument similar to that used in the previous sections. Setting =u−v, we get, subtracting (43) from (40), that satises
t+ [f]x=xx; (47a)
(x;0) =(x); (47b)
where
[f]≡f(+v)−f(v): (48)
Multiplying (47a) by (1 +t), where ¿1
2, and integrating the result on R×[0; T], we obtain,
after a few integrations by parts
(1 +T)k(·; T)k2
L2(R)+ 2
Z T
0
(1 +t)kx(·; t)k2
L2(R)dt
=k(·;0)k2
L2(R)+
Z T
0
(1 +t)−1k(·; t)k2
L2(R)dt
−2 Z T
0
(1 +t) Z +∞
−∞
(x; t)[f]xdxdt: (49)
Since
Z +∞
−∞
(x; t)[f]xdx= Z +∞
−∞
(x; t)(f′(+v)(x+vx)−f
′
(v)vx) dx
= Z +∞
−∞
(x; t)[f′]vxdx+ Z +∞
−∞
xf
′
(+v) dx;
we get, using (10),
Z +∞
−∞
(x; t)[f]xdx=a
Z +∞
−∞
so that we can write (49) as
As in the previous sections, we use the Sobolev inequality
k(·; t)kL2(R)6Ck(·; t)k2=3
L1(R)kx(·; t)k1L=23(R);
together with (42) and (45) to get
Z T
Proceeding as in (29) and (30), we then immediately get (46) and the following result.
Theorem 3. Let v(x; t) be the rarefaction wave (43) corresponding to an initial; expansive prole v0∈B(v−; v+);and let u0=v0+ where the disturbance is a bounded; integrable function. Then
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