17-21.2010. 130KB Jun 04 2011 12:07:40 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 21/2010
pp. 161-170
ABOUT THE EXISTENCE AND UNIQUENESS OF SOLUTION TO
FRACTIONAL BURGERS EQUATION
Amar Guesmia and Noureddine Daili
Abstract. In this work, we study local and global solutions of an evolution
problem governed by fractional B¨
urgers equations. We have generalized B¨
urgers
equation with a fractional degree of Laplacian in the main part and an algebraic
degree in nonlinear part. Such equations intervene, naturally, in continuum mechanics area. Our results prove existence, uniqueness and regularity of solutions of
Cauchy’s problem for Fractional B¨
urgers equation. These problems arise in a variety
of engineering analysis and design situations.
2000 Mathematics Subject Classification: Primary 26A33, 35Q53; Secondary
35M05, 35A35.
1.Introduction
The one-dimensional B¨
urgers equation
∂2
∂u
∂
u(t, x) =
u(t, x) + u(t, x)
2
∂t
∂x
∂x
(1.1)
was proposed by B¨
urgers ([3], 1948) in 1948 as a model for turbulent phenomena of
viscous fluids. Since then, B¨
urgers equation has been investigated in many fields of
application, such as traffic flows and formation of large clusters in the universe.
In order to model solutions of Navier-Stokes equations, several authors have
studied B¨
urgers equations with random initial conditions, including white and stable
noises.
Equation (1.1) can be solved in closed form (in terms of the initial conditions)
by using the Hopf-Cole substitution, which reduces it to a heat equation.
B¨
urgers equations involving in their linear parts fractional powers ∆α := −(−∆)α/2
of the Laplacian, α ∈ (0, 2], have been investigated in connection with certain models of hydrodynamical phenomena ; see Shlesinger and al. ([12], 1995), Funaki and
al. ([4], 1995) and Biler and al. ([2], 1998). In Biler and al. ([2]),
161
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Biller, Funaki and Woyczynski studied existence, uniqueness, regularity and
asymptotic behavior of solutions to the multidimensional fractal B¨
urgers-type equation
∂
u(t, x) = ν∆α u(t, x) − a∇ur (t, x)
(1.2)
∂t
where x ∈ Rd , d ≥ 1, α ∈ (0, 2], r ≥ 1, and a ∈ Rd . For α > 3/2 and d = 1 they
prove existence of a unique regular weak solution to (1.2) for initial conditions in
H 1 (R).
B¨
urgers equations in financial mathematics arise in connection with the behavior
of the risk premium of the market portfolio of risky assets under Black-Scholes
assumptions.
In this work, we study local and global solutions of an evolution problem governed by equations of fractional B¨
urgers kind. Namely, we study the time-fractional
B¨
urgers equation. We have generalized B¨
urgers equation with fractional degree of
Laplacian in the main part and algebraic degree in nonlinear part. Such equations
intervene, naturally, in continuum mechanics area and engineering mechanics. These
problems arise in a variety of engineering analysis and design situations.
Our results prove existence, uniqueness and regularity of solutions of Cauchy’s
problem to the following time-B¨
urgers equation :
1
ut = uxx − (u2 )x + f (x, t),
2
where
x ∈ I ⊂ R, t ≥ 0, u : I × R+ → R.
2.Mains Results
2.1. A Direct Approach to Weak Solutions
We study existence and uniqueness solutions of Cauchy’s problem. We generalize the following B¨
urgers equation
1
ut = uxx − (u2 )x + f (x, t)
2
for one fractional degree of Laplacian in a main part and one algebraic degree in
nonlinear part. We introduce and develop the following generalization:
1
ut = −Dα u − (u2 )x + f (x, t), 0 < α ≤ 2,
2
162
(2.1)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
with
u(x, 0) = u0 (x),
(2.2)
α
2
and Dα ≡ (−∂ 2 /∂x2 ) .
Using a priori elementary estimates, we prove results for Cauchy’s problem (2.1).
In particular, this will prove a role of operator −Dα and its power relative to the
nonlinear term uux .
Define Dα as ([9] and [10]):
Z x
1
α
(x − z)−α−1 v(z)dz,
(2.3)
(D v)(x) =
Γ(−α) 0
R∞
where Γ(α) = 0 z α−1 e−z dz denotes Euler’s Gamma function.
We look for weak solutions of problem (2.1) with initial data u(x, 0) = u0 (x) in
V2 such that
V2 = L∞ (]0, T [ ; L2 (I)) ∩ L2 (]0, T [ ; H 1 (I))
satisfying the identity
R
RtR
u(x, t)φ(x, t)dx − 0 u(x, t)φt (x, t)dxdt+
RtR
0
=
R
α
α
D 2 u(x, t)D 2 φ(x, t)dxdt −
u0 (x)φ(x, 0)dx +
RtR
0
RtR
0
1 2
2 u (x, t)φx (x, t)dxdt
(2.4)
f (x, t)dxdt,
for t ∈ ]0, T [ and φ(x, t) ∈ H 1 (I × ]0, T [).
In order to simplify our construction, suppose u(t) ∈ H 1 (I) for t ∈ ]0, T [
α
instead of u(t) ∈ H 2 (I) for t ∈ ]0, T [ which will can be waiting for an ordinary
generalization of a definition of a weak solution of one parabolic (see [7]).
Suppose, also, initial condition u0 (x) ∈ H 1 (I).
Theorem 1. ([2]) Let 32 < α ≤ 2, T > 0, and u0 (x) ∈ H 1 (I). Then Cauchy’s
problem (2.1)-(2.2) has an unique weak solution u ∈ V2 . Moreover, u satisfies the
following regularity properties:
α
u ∈ L∞ (]0, T [ ; H 1 (I)) ∩ L2 (]0, T [ ; H 1+ 2 (I))
and
α
ut ∈ L∞ (]0, T [ ; L2 (I)) ∩ L2 (]0, T [ ; H 2 (I))
163
(2.5)
(2.6)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Proof. Suppose u is a weak solution of (2.1)-(2.2) and let Sn be a truncation operator such that un = Sn u then we can consider the following approximate
problem:
1
(un )t = −Dα un − (u2n )x + f (x, t), 0 < α ≤ 2
(2.7)
2
with initial data un|t=0 = Sn u0 .
Let us multiply (2.7) by un , then
Z
Z
Z
Z
α
d
2
2
un (x, t) + (D 2 un (x, t)) + un (x, t)(un )x (x, t)un (x, t) = f (x, t)un (x, t)
dt
which implies
Z
Z
Z
Z
α
d
2
2
2
2
un (x, t) + (D un (x, t)) + un (x, t)(un )x (x, t) = f (x, t)un (x, t).
dt
One has
Z
u2n (x, t)(un )x (x, t)
Then it holds that
1 3
= ( un (x, t)) = CS(t) = C1 |un |2 .
3
I
2
α
d
|un |22 + D 2 un (t) ≤ (|f |2 + C1 ) |un |2 .
dt
2
(2.8)
Likewise, upon differentiation in formula (2.7) according to x and multiply by
(un )x , we obtain
R
R
α
d
u
(x,
t)(u
)
(x,
t)
+
(D 2 un (x, t))(un )x (x, t)
n
n
x
dt
+
which implies
R
1 d
2 dt
+
It holds that
R
1 2
2 (un (x, t))x (un )x (x, t)
R
=
R
f (x, t)(un )x (x, t),
R
α
u2n (x, t) + (D 2 un (x, t))(un )x (x, t)
1 2
2 (un (x, t))x (un )x (x, t)
=
R
f (x, t)(un )x (x, t)
2
α
d
|(un )x |22 + 2 D1+ 2 un (t) ≤ |(un )x |33 + 2 |f |2 |un |2
dt
2
164
(2.9)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
because
Z
Z
Z
Z
1
1
1 2
(un )x ux = un (un )x (un )xx =
un ((un )2x )x = −
(un )3x .
−
2
2
2
Now, a part of right member of (2.9) can be approximated by
7/(2+α)
3−7/(2+α)
|(un )x |33 ≤ kun k31,3 ≤ C kun k1+α/2 |un |2
≤ kun k21+α/2 + C |un |m
2 ,
for any m > 0.
Assumption α > 3/2 has been used in the interpolation of W 1,3 of norm of u by
norms of its fractional derivative 7/(2 + α). Indeed, that one follows from ([6], 1982;
p. 99). Devising this with (2.7), (2.8) and (2.9) we obtain
d
kun k21 + kun k21+α/2 ≤ C(|f |2 |un |2 + |un |22 + |un |m
2 + C1 )
dt
and by (2.8) one has
d
|un |22 ≤ (|f |2 + C1 ) |un |2 ⇒ |un (t)|2 ≤ M + |(un )0 |2 , ∀ t ∈ [0, T ],
dt
hence, we obtain
kun (t)k21
+
Z
0
t
kun (t)k21+α/2 ds ≤ C = C(T, f, k(un )0 k1 ).
(2.10)
Now, approximate a derivative according to the time of a solution. Multiply
(2.1) by (un )t
R
R
α
d
un (x, t)(un )t (x, t) + (D 2 un (x, t))(un )t (x, t)
dt
Hence
R
R
+ un (x, t)(un )x (x, t)(un )t (x, t) = f (x, t)(un )t (x, t).
1 d
2 dt
R
R
(u2n (x, t))t +
1
2
R
α
D 2 (u2n (x, t))t +
un (x, t)(un )x (x, t)(un )t (x, t) =
R
f (x, t)(un )t (x, t).
After some calculations, we obtain
Z
Z
α
2
d
2
2
2
|(un )t |2 + D (un )t = − (un )x (un )t + 2 f (x, t)(un )t (x, t)
dt
2
165
(2.11)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
since
Z
Z
Z
Z
1
1
un ((un )2t )x −
(un )x (un )2t .
− (un (un )x )t (un )t = − (un )x (un )2t −
2
2
Now, approximate a right member (2.7) by
1
2
R
1/α
≤
and
2−1/α
|(un )x | (un )2t ≤ C k(un )t kα/2 |(un )t |2
Z
1
2
|(un )x |2
k(un )t k21+α/2 + C |(un )t |22
f (x, t)(un )t (x, t) ≤ |f |2 |(un )t |2 .
A classical Gronwall inequality gives
Z t
2
k(un )t (s)k2α/2 ds ≤ C(T )
|(un )t (t)|2 +
(2.12)
0
It holds, from (2.10) and (2.12), that a solution un is bounded. Then it is sufficient in order to apply approximation Galerkin’s procedure. Hence, we can extract a
α
subsequence which converges to a limit u in L∞ (]0, T [ ; H 1 (I))∩L2 (]0, T [ ; H 1+ 2 (I)).
To finish, it remains to know if u is a solution of problem?
Since injection of H 1 (I) into L2 (I) is compact, we can apply Ascoli theorem and
conclude a strongly convergence of (un )n∈N to u in L2 (]0, T [ ; L2 (I)).
In order to conclude, it is enough to prove that (un )2 converges strongly to u2
in L1 (]0, T [ ; L2 (I)). Remark that
(un )2 − u2
1
≤ kun − ukL1 (]0, T [ ; L4 (I)) (kun kL1 (]0, T [ ; L4 (I))
L (]0, T [ ; L2 (I))
+ kukL1 (]0, T [ ; L4 (I)) ),
it is enough to prove that un − u converges strongly in L1 (]0, T [ ; L4 (I)). This
last result holds by Gagliardo-Nirenberg’s inequality ([1], [2])
1− 1
1
4
k∇(un − u)kL4 2 (]0, T [ ; L4 (I))
kun − ukL2 (]0, T [; L4 (I)) ≤ C kun − ukL2 (]0,
T [ ; L4 (I))
1− 1
4
≤ C kun − ukL2 (]0,
T [ ; L4 (I))
and to prove that Dα un converges strongly to Dα u in L1 (]0, T [ ; L2 (I)).
166
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
In the same way, we remark that
kDα un − Dα ukL1 (]0, T [ ; L2 (I)) ≤
2
∂ un
∂x2 −
2
∂2u
(
∂∂xu2n
1
α
∂x2
L1 (]0, T [ ; H 1+ α
2 (I))
L (]0, T [ ; H 1+ 2 (I))
2
+
∂∂xu2
α
L1 (]0, T [ ; H 1+ 2 (I))
)
and since the term ∂ 2 /∂x2 is linear, approach problem converges weakly to a
limit point, then the existence holds.
Now we prove uniqueness solution. Consider two weak solutions u and v of (2.1).
Then their difference w = u − v satisfies
α
2
R
2
2
d
D
w(t)
|w|
+
2
= 2 (vvx − u ux )w
2
dt
2
(2.13)
R
R 2
2
= −2 (vwwx − w ux ) = 2 w (vx /2 − ux ).
A right member of (2.13) can be limited and we obtain
1/α
2−1/α
|w|24 |vx − 2 ux |2 ≤ C kwkα/2 |w|2
≤
1
2
(|ux |2 + |vx |2 )
kwk21+α/2 + C |w|22 .
From (2.10), a factor (|ux |2 + |vx |2 ) is bounded. By Gronwall’s lemma it holds
that w(t) ≡ 0 on [0, T ]
2.2. Parabolic Reguralization
• For α > 3/2, a diffusion operator Dα is strong in order to control a nonlinear
part 21 (u2 )x , furthermore Cauchy problem (2.1) has one and only one solution.
• For α ≤ 3/2, we cannot wait to prove uniqueness of weak solution according
to the time for initial data (2.2). We shall use an other technical to obtain weak
solutions. The construction will be done by a parabolic regularization method.
Namely, we study the problem
ut = −Dα u − 21 (u2 )x + εuxx + f (x, t)
(2.14)
u(x, 0) = u0 (x),
with uε = u, ε > 0 (see, for example, ([1], 1979) ; ([8], 1969)).
In particular, solutions of B¨
urgers equation can be obtained as limits of solutions
of
1
ut = − (u2 )x + f (u) + εuxx as ε → 0.
2
167
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Theorem 2. ([2]) Let 0 < α ≤ 2. Let uε = u, (ε > 0), be a solution of Cauchy
problem (2.14), with u0 ∈ L1 ∩ H 1 , (u0 )x ∈ L1 . Then, for all t ≥ 0, we have
|u(t)|2 ≤ C2 ,
|u(t)|1 ≤ C1 ,
|ux (t)|1 ≤ Cx,1 .
Proof. The existence of solutions for regularized equation (2.14) is standard as
previously. Denote the operator A = −Dα − D2 . Then for every v ∈ D(A) (domain
of A)([1]), one has
Z
(Av)sgn(v(x)) ≤ 0
In fact,
and
Z
R
(Av)sgn(v(x)) = −
Z
(2.15)
(Dα v + D2 v)sgn(v(x))
R
(Av)sgn(v(x)) = lim s−1 (esA v(x) − v(x))sgn(v(x))
(s→0)
R
R
≤ lim sup s−1 ( ( esA v(x) − |v(x))|) ≤ 0
(s→0)
Then, let us multiply (2.14) by sgn(u) and integrate on I, we obtain
R
R α
R 1 2
d
|u(x,
t)|
=
−
(D
u
+
εu
)sgn(u)
−
xx
dt
2 (u )x sgn(u)
R
R
R
+ f (x, t)sgn(u) ≤ − 12 (u2 )x sgn(u) + f (x, t)sgn(u).
d
What allows to conclude that dt
|u(t)|1 is bounded. By integration, on [0, t], of this
last quantity we prove that |u(t)|1 is bounded.
Introduce a function sgnη called function of sign ([5]) of increasing regularization,
η > 0, such that sgnη → sgn as η → 0. For such regularization we have
R
R 2
(u )x sgnη u = (u2 )sgnη u I − u2 (sgn′η u)ux
R
R
= trace((u2 )sgnη u) I − u2 (sgn′η u)ux = MT r − u2 (sgn′η u)ux .
R
Therefore ux is bounded in H 1 for every ε > 0. We see that the integral
u2 (sgn′η u)ux converges to 0 as η → 0.
Then by multiplying (2.14) by sgn(ux ) we obtain
Z
Z
d
|ux |1 ≤ − (u ux )x sgn(ux ) + f (x, t)sgn(ux ).
dt
168
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Still, let us approach sgn by functions sgnη , we transform the integral
Z
Z
(u ux )x sgn(ux ) = trace((u ux )sgnη ux )|I − u uxx ux (sgn′η (ux )).
R
We see that the integral u uxx ux sgn′η (ux ) converges to 0 as η → 0. We can
now pass to a limit on ε when ε → 0 in regularized equation (2.14).
In what follows, by a weak solution of (2.1) we hear u ∈ L∞ ((0, T ) ; L2 (I)) and
satisfying the following equation:
R
RtR
RtR
u(x, t)φ(x, t) − 0 u(x, t)φt (x, t) + 0 (uDα φ − 21 u2 φx
=
R
u0 (x)φ(x, 0) +
RtR
0
f (x, t) for t ∈ (0, T ),
φ ∈ C ∞ (I × [0, T ]) with compact support. Let us note that we do not assume
u(t) ∈ H α/2 .
Corolary 1. Let 0 < α < 2, u0 ∈ L1 ∩ H 1 with (u0 )x ∈ L1 , there is a weak
solution u of (2.1) obtained as a limit of a sequence uε such that
u ∈ L∞ ((0, ∞) ; L∞ (I)) ∩ L∞ ((0, ∞) ; H 1/2−δ (I)),
for every δ > 0. Furthermore, u ∈ L∞ ((0, ∞) ; BV (I)) with
ku(t)kBV (I) ≤ |(u0 )x |1
Proof. From injection W 1,1 ⊂ H 1/2−δ we conclude that a subsequence uε converges weakly to a limit function u in L∞ ((0, ∞) ; H 1/2−δ (I)). A sequence uε is
bounded in L∞ holds from obvious inequality |u|∞ ≤ |ux |1 . The strong convergence
in L∞ ((0, ∞) ; H 1/2−δ (I)) is a consequence of Aubin-Lions’s lemma ([8], p. 57).
Remark 1.
• If α > 1/2 then weak solutions of (2.14) (constructed by a parabolic regularization method) remain in H 1 (R) for t ∈ [0, T ) for some T > 0. Moreover, if ku0 k1
is enough small, then these regulary solutions are global in the time.
• If α < 1 these weak solutions, defined on time-finite interval, introduce shocks.
References
[1] C. Bardos, P. Penel, U. Frisch and P.L. Sulem, Modified dissipativity for
a nonlinear evolution equation arising in turbulence, Arch. Rat. Mech. Anal.
71 (1979), 237-256.
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[2] P. Biler, T. Funaki and W.A. Woyczynski, Fractal B¨
urgers Equations, Journal
of differential equations 148 (1998), 9-46.
[3] J.M. B¨
urgers, A mathematical model illustrating the theory of turbulence,
Adv. Appl. Mech. 1 (1948), 171-199.
[4] T. Funaki, D. Surgailis and W. Woyczynski, Gibbs-Cox random fields and
B¨
urgers turbulence, Ann. Appl. Prob. 5 (1995), 701-735.
[5] A. Guesmia and N. Daili, Existence and Uniqueness of an Entropy Solution
for B¨
urgers Equations, Applied Mathematical Sciences (AMS), Vol. 2, 33 (2008),
1635-1664.
[6] D.B. Henry, How to remember the Sobolev inequalities, in “Differential Equations, Sao˜Paulo, 1981” (D.G.de Figueiredo and C.S. Ho Nig, Eds.), Lecture Notes
in Mathematics, N◦ 957, Springer, Berlin, (1982), 97-109.
[7] C. Hirsch, Numerical computation of internal and external flows. Volume
1: Fundamentals of numerical discretization, Wiley Series in Numerical Methods in
Engineering. Wiley-Interscience Publication, 1988.
[8] J.-L. Lions, Quelques m´ethodes de r´esolution des probl`emes aux limites non
lin´eaires, Dunod, Paris, 1969.
[9] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego,
1999.
[10] I. Podlubny, Fractional differential Equations, Ser. Math. Science and
Engineering, Acad. Press, San Diego, 1999.
[11] J.C. Saut, Sur quelques g´en´eralisations de l’´equation de Korteweg de Vries,
J. Math. Pures Appl. 58 (1979), 21-61.
[12] M.F. Shlesinger, G.M. Zaslavsky and U. Frisch (Eds.), L´evy Flights and related topics in physics, Vol. 450, Lecture Notes in Physiscs, Springer-Verlag, Berlin,
1995.
Amar Guesmia
Department of Mathematics
University of 20 Aoˆ
ut 1955
Route d’El-Hadaiek, B.P. 26-Skikda-21000, Algeria
email: guesmiasaid@yahoo.fr
Noureddine Daili
Department of Mathematics
7650, rue Querbes, # 15, Montr´eal, Qu´ebec, H3N 2B6, Canada
email: ijfnms@hotmail.fr
170
ISSN: 1582-5329
No. 21/2010
pp. 161-170
ABOUT THE EXISTENCE AND UNIQUENESS OF SOLUTION TO
FRACTIONAL BURGERS EQUATION
Amar Guesmia and Noureddine Daili
Abstract. In this work, we study local and global solutions of an evolution
problem governed by fractional B¨
urgers equations. We have generalized B¨
urgers
equation with a fractional degree of Laplacian in the main part and an algebraic
degree in nonlinear part. Such equations intervene, naturally, in continuum mechanics area. Our results prove existence, uniqueness and regularity of solutions of
Cauchy’s problem for Fractional B¨
urgers equation. These problems arise in a variety
of engineering analysis and design situations.
2000 Mathematics Subject Classification: Primary 26A33, 35Q53; Secondary
35M05, 35A35.
1.Introduction
The one-dimensional B¨
urgers equation
∂2
∂u
∂
u(t, x) =
u(t, x) + u(t, x)
2
∂t
∂x
∂x
(1.1)
was proposed by B¨
urgers ([3], 1948) in 1948 as a model for turbulent phenomena of
viscous fluids. Since then, B¨
urgers equation has been investigated in many fields of
application, such as traffic flows and formation of large clusters in the universe.
In order to model solutions of Navier-Stokes equations, several authors have
studied B¨
urgers equations with random initial conditions, including white and stable
noises.
Equation (1.1) can be solved in closed form (in terms of the initial conditions)
by using the Hopf-Cole substitution, which reduces it to a heat equation.
B¨
urgers equations involving in their linear parts fractional powers ∆α := −(−∆)α/2
of the Laplacian, α ∈ (0, 2], have been investigated in connection with certain models of hydrodynamical phenomena ; see Shlesinger and al. ([12], 1995), Funaki and
al. ([4], 1995) and Biler and al. ([2], 1998). In Biler and al. ([2]),
161
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Biller, Funaki and Woyczynski studied existence, uniqueness, regularity and
asymptotic behavior of solutions to the multidimensional fractal B¨
urgers-type equation
∂
u(t, x) = ν∆α u(t, x) − a∇ur (t, x)
(1.2)
∂t
where x ∈ Rd , d ≥ 1, α ∈ (0, 2], r ≥ 1, and a ∈ Rd . For α > 3/2 and d = 1 they
prove existence of a unique regular weak solution to (1.2) for initial conditions in
H 1 (R).
B¨
urgers equations in financial mathematics arise in connection with the behavior
of the risk premium of the market portfolio of risky assets under Black-Scholes
assumptions.
In this work, we study local and global solutions of an evolution problem governed by equations of fractional B¨
urgers kind. Namely, we study the time-fractional
B¨
urgers equation. We have generalized B¨
urgers equation with fractional degree of
Laplacian in the main part and algebraic degree in nonlinear part. Such equations
intervene, naturally, in continuum mechanics area and engineering mechanics. These
problems arise in a variety of engineering analysis and design situations.
Our results prove existence, uniqueness and regularity of solutions of Cauchy’s
problem to the following time-B¨
urgers equation :
1
ut = uxx − (u2 )x + f (x, t),
2
where
x ∈ I ⊂ R, t ≥ 0, u : I × R+ → R.
2.Mains Results
2.1. A Direct Approach to Weak Solutions
We study existence and uniqueness solutions of Cauchy’s problem. We generalize the following B¨
urgers equation
1
ut = uxx − (u2 )x + f (x, t)
2
for one fractional degree of Laplacian in a main part and one algebraic degree in
nonlinear part. We introduce and develop the following generalization:
1
ut = −Dα u − (u2 )x + f (x, t), 0 < α ≤ 2,
2
162
(2.1)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
with
u(x, 0) = u0 (x),
(2.2)
α
2
and Dα ≡ (−∂ 2 /∂x2 ) .
Using a priori elementary estimates, we prove results for Cauchy’s problem (2.1).
In particular, this will prove a role of operator −Dα and its power relative to the
nonlinear term uux .
Define Dα as ([9] and [10]):
Z x
1
α
(x − z)−α−1 v(z)dz,
(2.3)
(D v)(x) =
Γ(−α) 0
R∞
where Γ(α) = 0 z α−1 e−z dz denotes Euler’s Gamma function.
We look for weak solutions of problem (2.1) with initial data u(x, 0) = u0 (x) in
V2 such that
V2 = L∞ (]0, T [ ; L2 (I)) ∩ L2 (]0, T [ ; H 1 (I))
satisfying the identity
R
RtR
u(x, t)φ(x, t)dx − 0 u(x, t)φt (x, t)dxdt+
RtR
0
=
R
α
α
D 2 u(x, t)D 2 φ(x, t)dxdt −
u0 (x)φ(x, 0)dx +
RtR
0
RtR
0
1 2
2 u (x, t)φx (x, t)dxdt
(2.4)
f (x, t)dxdt,
for t ∈ ]0, T [ and φ(x, t) ∈ H 1 (I × ]0, T [).
In order to simplify our construction, suppose u(t) ∈ H 1 (I) for t ∈ ]0, T [
α
instead of u(t) ∈ H 2 (I) for t ∈ ]0, T [ which will can be waiting for an ordinary
generalization of a definition of a weak solution of one parabolic (see [7]).
Suppose, also, initial condition u0 (x) ∈ H 1 (I).
Theorem 1. ([2]) Let 32 < α ≤ 2, T > 0, and u0 (x) ∈ H 1 (I). Then Cauchy’s
problem (2.1)-(2.2) has an unique weak solution u ∈ V2 . Moreover, u satisfies the
following regularity properties:
α
u ∈ L∞ (]0, T [ ; H 1 (I)) ∩ L2 (]0, T [ ; H 1+ 2 (I))
and
α
ut ∈ L∞ (]0, T [ ; L2 (I)) ∩ L2 (]0, T [ ; H 2 (I))
163
(2.5)
(2.6)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Proof. Suppose u is a weak solution of (2.1)-(2.2) and let Sn be a truncation operator such that un = Sn u then we can consider the following approximate
problem:
1
(un )t = −Dα un − (u2n )x + f (x, t), 0 < α ≤ 2
(2.7)
2
with initial data un|t=0 = Sn u0 .
Let us multiply (2.7) by un , then
Z
Z
Z
Z
α
d
2
2
un (x, t) + (D 2 un (x, t)) + un (x, t)(un )x (x, t)un (x, t) = f (x, t)un (x, t)
dt
which implies
Z
Z
Z
Z
α
d
2
2
2
2
un (x, t) + (D un (x, t)) + un (x, t)(un )x (x, t) = f (x, t)un (x, t).
dt
One has
Z
u2n (x, t)(un )x (x, t)
Then it holds that
1 3
= ( un (x, t)) = CS(t) = C1 |un |2 .
3
I
2
α
d
|un |22 + D 2 un (t) ≤ (|f |2 + C1 ) |un |2 .
dt
2
(2.8)
Likewise, upon differentiation in formula (2.7) according to x and multiply by
(un )x , we obtain
R
R
α
d
u
(x,
t)(u
)
(x,
t)
+
(D 2 un (x, t))(un )x (x, t)
n
n
x
dt
+
which implies
R
1 d
2 dt
+
It holds that
R
1 2
2 (un (x, t))x (un )x (x, t)
R
=
R
f (x, t)(un )x (x, t),
R
α
u2n (x, t) + (D 2 un (x, t))(un )x (x, t)
1 2
2 (un (x, t))x (un )x (x, t)
=
R
f (x, t)(un )x (x, t)
2
α
d
|(un )x |22 + 2 D1+ 2 un (t) ≤ |(un )x |33 + 2 |f |2 |un |2
dt
2
164
(2.9)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
because
Z
Z
Z
Z
1
1
1 2
(un )x ux = un (un )x (un )xx =
un ((un )2x )x = −
(un )3x .
−
2
2
2
Now, a part of right member of (2.9) can be approximated by
7/(2+α)
3−7/(2+α)
|(un )x |33 ≤ kun k31,3 ≤ C kun k1+α/2 |un |2
≤ kun k21+α/2 + C |un |m
2 ,
for any m > 0.
Assumption α > 3/2 has been used in the interpolation of W 1,3 of norm of u by
norms of its fractional derivative 7/(2 + α). Indeed, that one follows from ([6], 1982;
p. 99). Devising this with (2.7), (2.8) and (2.9) we obtain
d
kun k21 + kun k21+α/2 ≤ C(|f |2 |un |2 + |un |22 + |un |m
2 + C1 )
dt
and by (2.8) one has
d
|un |22 ≤ (|f |2 + C1 ) |un |2 ⇒ |un (t)|2 ≤ M + |(un )0 |2 , ∀ t ∈ [0, T ],
dt
hence, we obtain
kun (t)k21
+
Z
0
t
kun (t)k21+α/2 ds ≤ C = C(T, f, k(un )0 k1 ).
(2.10)
Now, approximate a derivative according to the time of a solution. Multiply
(2.1) by (un )t
R
R
α
d
un (x, t)(un )t (x, t) + (D 2 un (x, t))(un )t (x, t)
dt
Hence
R
R
+ un (x, t)(un )x (x, t)(un )t (x, t) = f (x, t)(un )t (x, t).
1 d
2 dt
R
R
(u2n (x, t))t +
1
2
R
α
D 2 (u2n (x, t))t +
un (x, t)(un )x (x, t)(un )t (x, t) =
R
f (x, t)(un )t (x, t).
After some calculations, we obtain
Z
Z
α
2
d
2
2
2
|(un )t |2 + D (un )t = − (un )x (un )t + 2 f (x, t)(un )t (x, t)
dt
2
165
(2.11)
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
since
Z
Z
Z
Z
1
1
un ((un )2t )x −
(un )x (un )2t .
− (un (un )x )t (un )t = − (un )x (un )2t −
2
2
Now, approximate a right member (2.7) by
1
2
R
1/α
≤
and
2−1/α
|(un )x | (un )2t ≤ C k(un )t kα/2 |(un )t |2
Z
1
2
|(un )x |2
k(un )t k21+α/2 + C |(un )t |22
f (x, t)(un )t (x, t) ≤ |f |2 |(un )t |2 .
A classical Gronwall inequality gives
Z t
2
k(un )t (s)k2α/2 ds ≤ C(T )
|(un )t (t)|2 +
(2.12)
0
It holds, from (2.10) and (2.12), that a solution un is bounded. Then it is sufficient in order to apply approximation Galerkin’s procedure. Hence, we can extract a
α
subsequence which converges to a limit u in L∞ (]0, T [ ; H 1 (I))∩L2 (]0, T [ ; H 1+ 2 (I)).
To finish, it remains to know if u is a solution of problem?
Since injection of H 1 (I) into L2 (I) is compact, we can apply Ascoli theorem and
conclude a strongly convergence of (un )n∈N to u in L2 (]0, T [ ; L2 (I)).
In order to conclude, it is enough to prove that (un )2 converges strongly to u2
in L1 (]0, T [ ; L2 (I)). Remark that
(un )2 − u2
1
≤ kun − ukL1 (]0, T [ ; L4 (I)) (kun kL1 (]0, T [ ; L4 (I))
L (]0, T [ ; L2 (I))
+ kukL1 (]0, T [ ; L4 (I)) ),
it is enough to prove that un − u converges strongly in L1 (]0, T [ ; L4 (I)). This
last result holds by Gagliardo-Nirenberg’s inequality ([1], [2])
1− 1
1
4
k∇(un − u)kL4 2 (]0, T [ ; L4 (I))
kun − ukL2 (]0, T [; L4 (I)) ≤ C kun − ukL2 (]0,
T [ ; L4 (I))
1− 1
4
≤ C kun − ukL2 (]0,
T [ ; L4 (I))
and to prove that Dα un converges strongly to Dα u in L1 (]0, T [ ; L2 (I)).
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A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
In the same way, we remark that
kDα un − Dα ukL1 (]0, T [ ; L2 (I)) ≤
2
∂ un
∂x2 −
2
∂2u
(
∂∂xu2n
1
α
∂x2
L1 (]0, T [ ; H 1+ α
2 (I))
L (]0, T [ ; H 1+ 2 (I))
2
+
∂∂xu2
α
L1 (]0, T [ ; H 1+ 2 (I))
)
and since the term ∂ 2 /∂x2 is linear, approach problem converges weakly to a
limit point, then the existence holds.
Now we prove uniqueness solution. Consider two weak solutions u and v of (2.1).
Then their difference w = u − v satisfies
α
2
R
2
2
d
D
w(t)
|w|
+
2
= 2 (vvx − u ux )w
2
dt
2
(2.13)
R
R 2
2
= −2 (vwwx − w ux ) = 2 w (vx /2 − ux ).
A right member of (2.13) can be limited and we obtain
1/α
2−1/α
|w|24 |vx − 2 ux |2 ≤ C kwkα/2 |w|2
≤
1
2
(|ux |2 + |vx |2 )
kwk21+α/2 + C |w|22 .
From (2.10), a factor (|ux |2 + |vx |2 ) is bounded. By Gronwall’s lemma it holds
that w(t) ≡ 0 on [0, T ]
2.2. Parabolic Reguralization
• For α > 3/2, a diffusion operator Dα is strong in order to control a nonlinear
part 21 (u2 )x , furthermore Cauchy problem (2.1) has one and only one solution.
• For α ≤ 3/2, we cannot wait to prove uniqueness of weak solution according
to the time for initial data (2.2). We shall use an other technical to obtain weak
solutions. The construction will be done by a parabolic regularization method.
Namely, we study the problem
ut = −Dα u − 21 (u2 )x + εuxx + f (x, t)
(2.14)
u(x, 0) = u0 (x),
with uε = u, ε > 0 (see, for example, ([1], 1979) ; ([8], 1969)).
In particular, solutions of B¨
urgers equation can be obtained as limits of solutions
of
1
ut = − (u2 )x + f (u) + εuxx as ε → 0.
2
167
A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Theorem 2. ([2]) Let 0 < α ≤ 2. Let uε = u, (ε > 0), be a solution of Cauchy
problem (2.14), with u0 ∈ L1 ∩ H 1 , (u0 )x ∈ L1 . Then, for all t ≥ 0, we have
|u(t)|2 ≤ C2 ,
|u(t)|1 ≤ C1 ,
|ux (t)|1 ≤ Cx,1 .
Proof. The existence of solutions for regularized equation (2.14) is standard as
previously. Denote the operator A = −Dα − D2 . Then for every v ∈ D(A) (domain
of A)([1]), one has
Z
(Av)sgn(v(x)) ≤ 0
In fact,
and
Z
R
(Av)sgn(v(x)) = −
Z
(2.15)
(Dα v + D2 v)sgn(v(x))
R
(Av)sgn(v(x)) = lim s−1 (esA v(x) − v(x))sgn(v(x))
(s→0)
R
R
≤ lim sup s−1 ( ( esA v(x) − |v(x))|) ≤ 0
(s→0)
Then, let us multiply (2.14) by sgn(u) and integrate on I, we obtain
R
R α
R 1 2
d
|u(x,
t)|
=
−
(D
u
+
εu
)sgn(u)
−
xx
dt
2 (u )x sgn(u)
R
R
R
+ f (x, t)sgn(u) ≤ − 12 (u2 )x sgn(u) + f (x, t)sgn(u).
d
What allows to conclude that dt
|u(t)|1 is bounded. By integration, on [0, t], of this
last quantity we prove that |u(t)|1 is bounded.
Introduce a function sgnη called function of sign ([5]) of increasing regularization,
η > 0, such that sgnη → sgn as η → 0. For such regularization we have
R
R 2
(u )x sgnη u = (u2 )sgnη u I − u2 (sgn′η u)ux
R
R
= trace((u2 )sgnη u) I − u2 (sgn′η u)ux = MT r − u2 (sgn′η u)ux .
R
Therefore ux is bounded in H 1 for every ε > 0. We see that the integral
u2 (sgn′η u)ux converges to 0 as η → 0.
Then by multiplying (2.14) by sgn(ux ) we obtain
Z
Z
d
|ux |1 ≤ − (u ux )x sgn(ux ) + f (x, t)sgn(ux ).
dt
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A. Guesmia, N. Daili - About the existence and uniqueness of solution to...
Still, let us approach sgn by functions sgnη , we transform the integral
Z
Z
(u ux )x sgn(ux ) = trace((u ux )sgnη ux )|I − u uxx ux (sgn′η (ux )).
R
We see that the integral u uxx ux sgn′η (ux ) converges to 0 as η → 0. We can
now pass to a limit on ε when ε → 0 in regularized equation (2.14).
In what follows, by a weak solution of (2.1) we hear u ∈ L∞ ((0, T ) ; L2 (I)) and
satisfying the following equation:
R
RtR
RtR
u(x, t)φ(x, t) − 0 u(x, t)φt (x, t) + 0 (uDα φ − 21 u2 φx
=
R
u0 (x)φ(x, 0) +
RtR
0
f (x, t) for t ∈ (0, T ),
φ ∈ C ∞ (I × [0, T ]) with compact support. Let us note that we do not assume
u(t) ∈ H α/2 .
Corolary 1. Let 0 < α < 2, u0 ∈ L1 ∩ H 1 with (u0 )x ∈ L1 , there is a weak
solution u of (2.1) obtained as a limit of a sequence uε such that
u ∈ L∞ ((0, ∞) ; L∞ (I)) ∩ L∞ ((0, ∞) ; H 1/2−δ (I)),
for every δ > 0. Furthermore, u ∈ L∞ ((0, ∞) ; BV (I)) with
ku(t)kBV (I) ≤ |(u0 )x |1
Proof. From injection W 1,1 ⊂ H 1/2−δ we conclude that a subsequence uε converges weakly to a limit function u in L∞ ((0, ∞) ; H 1/2−δ (I)). A sequence uε is
bounded in L∞ holds from obvious inequality |u|∞ ≤ |ux |1 . The strong convergence
in L∞ ((0, ∞) ; H 1/2−δ (I)) is a consequence of Aubin-Lions’s lemma ([8], p. 57).
Remark 1.
• If α > 1/2 then weak solutions of (2.14) (constructed by a parabolic regularization method) remain in H 1 (R) for t ∈ [0, T ) for some T > 0. Moreover, if ku0 k1
is enough small, then these regulary solutions are global in the time.
• If α < 1 these weak solutions, defined on time-finite interval, introduce shocks.
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Amar Guesmia
Department of Mathematics
University of 20 Aoˆ
ut 1955
Route d’El-Hadaiek, B.P. 26-Skikda-21000, Algeria
email: guesmiasaid@yahoo.fr
Noureddine Daili
Department of Mathematics
7650, rue Querbes, # 15, Montr´eal, Qu´ebec, H3N 2B6, Canada
email: ijfnms@hotmail.fr
170