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LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY CONDITION TO EULER EQUATIONS WITH DAMPING

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LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY CONDITION TO EULER EQUATIONS WITH DAMPING CHAO-JIANG XU AND TONG YANG Abstract In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in [13] to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood-Paley theory and justifies the transformation introduced in [13] in a rigorous setting. Key words Euler equations, physical vacuum boundary condition, Littlewood- Paley theory, local existence. A.M.S. Classification 35L67, 35L65, 35L05. 1. Introduction In this paper, we are interested in the time evolution of a gas connecting to vacuum with physical boundary condition. By assuming that the governed equations for the gas dynamics are Euler equations with linear damping, cf. [16] for physical interpretation, one can see that the system fails to be strictly hyperbolic at the vacuum boundary because the characteristics of different families coincide. As discussed in the previous works, cf. [5, 11, 12, 13], the canonical vacuum boundary behavior is the case when the space derivative of the enthalpy is bounded but not zero. In this case, the pressure has its non-zero finite effect on the evolution of the vacuum boundary.

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LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY
CONDITION TO EULER EQUATIONS WITH DAMPING
CHAO-JIANG XU AND TONG YANG
Abstract Inthispaper,weconsiderthelocalexistenceofsolutionstoEulerequations
withlineardampingundertheassumptionofphysicalvacuumboundarycondition. By
using the transformation introduced in [13] to capture the singularity of the boundary,
we prove a local existence theorem on a perturbation of a planar wave solution by
using Littlewood-Paley theory and justifies the transformation introduced in [13] in a
rigorous setting.
Key words Euler equations, physical vacuum boundary condition, Littlewood-
Paley theory, local existence.
A.M.S. Classification 35L67, 35L65, 35L05.
1. Introduction
In this paper, we are interested in the time evolution of a gas connecting to vacuum
withphysicalboundarycondition. Byassumingthatthegovernedequationsforthegas
dynamics are Euler equations with linear damping, cf. [16] for physical interpretation,
one can see that the system fails to be strictly hyperbolic at the vacuum boundary
because the characteristics of different families coincide. As discussed in the previous
works, cf. [5, 11, 12, 13], the canonical vacuum boundary behavior is the case when
the space derivative of the enthalpy is bounded but not zero. In this case, the pressure
has its non-zero finite effect on the evolution of the vacuum boundary. However, for
this canonical (physical) case, the system becomes singular in the sense that it can not
be symmetrizable with regular coefficients so that the local existence theory for the
classical hyperbolic systems can not be applied. Furthermore, the linearized equation
attheboundarygivesaKeyldishtypeequationforwhichgenerallocalexistencetheory
is still not known. Notice that this linearized equation is quite different from the one
considered in [18] for weakly hyperbolic equation which is of Tricomi type. To capture
this singularity in the nonlinear settting, a transformation was introduced in [13] and
some local existence results for bounded domain were also discussed. The transformed
equationisasecondordernonlinearwaveequationofanunknownfunction `(y;t)with
¡1coefficients as functions of y `(y;t) and `(0;t) · 0. Along the vacuum boundary,
the physical boundary condition implies that the coefficients are functions of ` (0;t)y
which are bounded and away from zero. Hence, the wave equation has no singularity
or degeneracy. However, its coefficients have the above special form so that the local
existencetheorydevelopedfortheclassicalnonlinearwaveequationcannotbeapplied
directly, [8, 9]. There are other works on this system with vacuum, please refer to
[6, 10] ect. and reference therein.
TO APPEAR IN “J. DIFF. EQU.”
12 XU AND YANG
Even though a transformation to capture the singularity in the physical boundary
condition at vacuum interface is introduced in [13], the energy method presented there
may not give a rigorous proof of the existence theory, especially in the general setting.
It is because the coefficients in the reduced wave equation which are power functions
¡1of y ` correspond to the fractional differentiations of `. Under this consideration,
we think the application of Littlewood-Paley theory based on Fourier theory is more
appropriate. Therefore, as the first step in this direction, in this paper we will study
the local existence of solutions satisfying the physical boundary condition when the
initial data is a small perturbation of a planar wave solution where the enthalpy is
linear in the space variable, [11]. By applying the Littlewood-Paley theory, we obtain
the solution local in time with the prescribed physical boundary condition.
Precisely,weconsidertheonedimensionalcompressibleEulerequationsforisentropic
flow with damping in Eulerian coordinates
‰ +(‰u) =0;t x
(1.1) ‰u +‰uu +p(‰) =¡‰u;t x x
where ‰, u and p(‰) are density, velocity and pressure respectively. And the linear
frictional coefficient is normalized to 1. When the initial density function contains
vacuum, the vacuum boundary Γ is defined as
Γ=clf(~x;t)j‰(~x;t)>0g\clf(~x;t)j‰(~x;t)=0g:
Since the second equation in (1.1) can be rewritten as
u +uu +i =¡u;t x x
with i being the enthalpy, one can see that the term i represents the effect of thex
pressure on the particle path, in particular, on the vacuum boundary. It is shown
in [12, 15, 19] that there is no global existence of regular solutions satisfying i · 0x
1along the vacuum boundary. That is, in general, i is not C crossing the vacuum
boundary. Hence, the canonical behavior of the vacuum boundary should satisfy the
condition i =0 and is bounded. This special feature of the solution can be illustratedx
by the stationary solutions and some self-similar solutions, also for different physical
systems,suchasEuler-PoissonequationsforgaseousstarsandNavier-Stokesequations,
cf. [5, 11, 13, 17]. Notice that the charateristics of Euler equations is u§ c, withp
2cc = p (‰). And for isentropic polytropy gas, i = , where ? > 1 is the adiabatic‰ ?¡1
constant. Hence the characteristics are singular with infinite space derivative at the
vacuum boundary if physical boundary condition is assumed. This singularity yields
the smooth reflection of the characteristic curves on the vacuum boundary and then
causes analytical difficulty.
Another way to view the canonical boundary condition comes from the study of
porous media equation. It is known that the Euler equations with linear damping
behave like the porous media equation at least away from vacuum when t ! 1,
cf.[7] and some corresponding results in the weak sense with v which will not
be discussed here. For the porous media equation, the free boundary of the support
of the solution has a canonical behavior which would be the same as or similar to
the one described above for Euler equations with damping. However, there is still
no satisfatory results on the change of solution behavior along the vacuum boundary
6LOCAL EXISTENCE 3
even though the corresponding waiting time problem for porous media equation is well
understood, cf.[1].
In this paper, we will concentrate on the Euler equations with linear damping when
the initial data is a small perturbation of a planar wave in one dimensional space.
Since our concern is the behavior of the solution related to vacuum and any shock
wave vanishes at vacuum [14], it is reasonable to consider our problem without shock
waves. In fact, any shock wave appears initially or in finite time will decay to zero
exponentially in time because of the dissipation from the linear damping. By using
the special property of the one dimensional gas dynamics, we can rewrite the system
(1.1) by using Lagrangian coordinates to make all the particle paths, in particular the
vacuum boundary, as straight lines. (1.1) in Lagrangian coordinates takes the form
v ¡u =0;t »
(1.2) u +p(v) =¡u;t »
Rx1where v = is the specific volume and » = ‰(y;t)dy. Moreover, we assume that the
‰ 0
2 ¡?pressurefunctionsatisfiesthe?-law, i.e.,p(v)=? v ,? > 1. Noticethatthephysical
singularity, i = 0 but bounded, along the vacuum boundary in Eulerian coordinatesx
corresponds to 0<jp (v)j<1 in the Lagrangian coordinates.»
In order to capture this singularity in the solution and symmetrize the system (1.2),
the following coordinate transformation was introduced in [13],
2?
?¡1» =y :
Here, we assume that the initial density function ‰ (x) = 0 for x < 0 in the Eulerian0
coordinates. Then the system (1.2) can be rewritten as
`(v) +„u¯ =0;t y
(1.3) u +„`¯ (v) =¡u; y > 0; t>0;t y
p
?¡12 ?? ¡
2where `(v)= v , and
?¡1
?+12 ?+1(?¡1)? ¡ ¡1?¡1 ?¡12„¯ = p (vy ) =•(y `) ;
?
for some positive constant •. Without any ambiguity and up to a scaling, • can be
chosen to be 1 and we still denote the independent variable by y for simplicity of
notation. Notice that near the vacuum boundary, both `(v) and „¯ are bounded awayy
from zero under the physical boundary condition.
Therefore, the vacuum problem considered can be formulated into the following
boundary value problem:
¡1 ¡1(1.4) („ w ) ¡(„w ) +„ w =0;t t y y t
(1.5) (w;w )j =(w ;w );t t=0 0 1
(1.6) w(0;t)=0;
¡1(1.7) 0<C •y w(y;t)•C ;1 2
¡1 fi 2‘with „ = (y w(y;t)) ;fi > 1; and compatibilities conditions @ w (0) = 0;‘ =0y
0;1;2;¢¢¢ :
64 XU AND YANG
It is easy to see that the above equation has a special linear unbounded solution for
y ‚ 0 given by w(y;t) = a y with constant a > 0. This solution is also obtained in0 0
[11] together with other self-similar solutions with physical boundary condition. To
justify the above transformation for local existence purpose, we will consider the local
existence of solution when the initial data is a small perturbation to the above special
solution.
That is, the initial data is assumed to be
w (y)=y(a +v (y)); w (y)=yv (y);0 0 0 1 1
for some constant a >0. And the solution is of following type0
w(y;t)=y(a +v(y;t)):0
Then the problem on (1.4)-(1.7) in this setting becomes
2fi+2 fiv2 3fi¡1 2 2 t(1.8) v ¡(„ v ) +fi(a +v) v ¡ „ v ¡ +v =0;tt y y 0 y ty
y a +v0
s s¡1(1.9) (v;v )j =(v ;v )2H (R )£H (R );t t=0 0 1 + +
(1.10) v(0;t)=0; t>0;
1
(1.11) kvk 1 • a ;L ([0;T]£ ) 0+ 2
fiwith „=(a +v(y;t)) and fi>1.0
For this problem, we have the following main theorem in this paper.
Theorem 1.1. Suppose that, for some b >0, we have0
1
(1.12) Suppv ; Suppv ‰[b ;+1[; kv k 1 • a ;0 1 0 0 L ( ) 0+ 4
3 ¡fiand s> . Then there exists 0<T <a b such that the problems (1.8)-(1.11) has a002
unique solution
s 0;1 s¡1v2C([0;T];H (R ))\C ([0;T];H (R )):+ +
Notice that the case when b = 0 is more difficult and will not be discussed here.0
Since the solution is regular up to the vacuum boundary and the density function in
positive except on the vacuum boundary, the result in Theorem 1.1 can be reduced
straightforwardly to the solution to the Equations (1.1).
Notice that here the initial perturbation is in a compact subset in (0;1) and the
local time existence is proved before the perturbation influence the propagation of the
boundary. Therefore, it is interesting and important to consider how the behavior
of the boundary changes in later time due to the perturbation. But this is not in
the scope of this paper and will be pursued by the authors in the future. For this,
the transformation introduced in [13] could still be useful. Furthermore, the physical
boundaryconditionholdsalsoformulti-dimensionalspacebyconsideringthestationary
solutions, [5]. Hence, the evolution of the vacuum interface in multi-dimensional space
can also be considered with more difficulty because there is no Lagrangian coordinates
to fix the vacuum interface.
The rest of the paper is arranged as follows. In Section 2, we shall briefly include
the Littlewood-Paley theory for the proof of local existence. The proof of Theorem
RRLOCAL EXISTENCE 5
1.1 is given in Section 3 where a linearized system is analyzed to yield a sequence of
solutions being convergent to the one in Theorem 1.1.
2. Littlewood-Paley theory
Inthissection,wewillrecallsomeelementarypropertiesofLittlewood-Paleytheory
for the Sobolev spaces, for the details please refer to [2, 3, 4]. Set
s d 0 2 s=2 2 dˆH (R )=ff 2S ;(1+j»j ) f 2L (R )g;
2 s=2ˆs 2with the norm kfk =k(1+j»j ) fk . We consider now a dyadic decompositionH L
dofR . For K >1 a fixed constant, and p2N , we set+
d ¡1 p p+1(2.1) C =f»2R ;K 2 •j»j•K2 g;p
d +1and C = B(0;K) =f» 2R ;j»j• Kg, then fC g is a uniformly finite recover of¡1 p ¡1
dR , that means, ifjp¡qj‚N =2(1+2log K)+2, we haveC \C =;.1 q p2
1 dWecanalsoconstructtwofunctions’;ˆ2C (R ),withSuppˆ‰C ;Supp’‰C ,¡1 00
dsuch that for any »2R and N ,0
1 N ¡10X X
¡p ¡p ¡N0ˆ(»)+ ’(2 »)=1; ˆ(»)+ ’(2 »)=ˆ(2 »):
p=0 p=0
Then one can define the following operators of localization in Fourier space, for
0 du2S (R ),
Z
¡1 ¡p pd pΔ u=u =F (’(2 ¢)uˆ(¢))=2 f(2 y)u(x¡y)dy; for p2Np p
d
and
¡1Δ u=u =F (ˆ(¢)uˆ(¢));¡1 ¡1
¡1whereuˆ =F(u)denotestheFouriertransformationofu,andf =F (’). ItisevidentP10 0 0that u 2S for any u2S , Suppuˆ ‰C , and u= u , in sense ofS .p p p pp=¡1
1Since Suppuˆ ‰C , Paley-Wienner-Schwartz theorem implies that u 2C and thep p p
Sobolev space can be characterized as follows.
Lemma 2.1. For s>0, the following properties are equivalent.
s d(a) u2H (R );P1 0 ¡ps 2(b) u= u inS , Supp uˆ ‰C andku k 2 •c 2 ; fc g2‘ ;p p p p L p pp=¡1P1 0 p ¡ps 2(c) u= u inS , Supp uˆ ‰B(0;K 2 ) andku k 2 •c 2 ; fc g2‘ ;p p 1 p L p pp=¡1P
1 0 1 d(d) u= u inS , u 2C and for any fi2N ;jfij• [s]+1,p pp=¡1
fi ¡p(s¡jfij) 2kD u k 2 •c 2 ; fc g 2‘ :p L p;fi p;fi p2
Remark : The equivalence of (a) and (b) holds for all s2R.
1For the L estimate, we need the following lemma.
1 d 1 dLemma 2.2. Suppose that a2 L (R );Supp aˆ‰ B(0;R), then a2 C (R ), and for
dany fi2N there exist C(d;fi)>0 such that
fi jfij
1 1(2.2) kD ak •C(d;fi)R kak :L L
NR6 XU AND YANG
¡N0For some N large enough, B(0;4K2 ) is a very small ball. Set0
0 ¡N0C =C +B(0;4K2 );00
0 p 0thenfC g=f2 C g has the same properties asfC g. We definepp 0
X X X
S u= u ; T v = (S u)v ; R(u;v)= u v :q p u q q p q
¡1•p•q¡N q0 jp¡qj<N0
Then, we have
uv =T v+T u+R(u;v);u v
and the following lemma.
1 s sLemma 2.3. (a), For any a 2 L , for any s 2 R, the maps T : H ! H isa
continuous and
s s 1(2.3) kT k •C kak :a L(H ;H ) s L
s s1 2(b) If u2H ;v2H ;s +s >0, we have1 2
s s(2.4) kR(u;v)k s +s ¡d=2 •Ckuk kvk :1 21 2 H HH
s d 1 s d 1(c) If s‚ 0, then H (R )\L is an algebra, and for any u;v 2 H (R )\L , we
have
s 1 s 1 s(2.5) kuvk •C(kuk kvk +kvk kuk );H L H L H
where C depends only on d;s.
A more general case of lemma 2.3 is the following.
1 1 s d 1Lemma2.4. LetF 2C (R );F(0)=0. Iff 2H (R )\L ;s‚0, isarealfunction,
s nthen the composition F(f)2H (R ) and
s 1 skF(f)k •C(F;s;kfk )kfk ;H L H
with
[s]+2X
j¡1(j)C(F;s;kfk 1)=C Sup jF (t)jkfk :1L d 0•t•kfk 1 LL
j=1
For later use, we also need the following estimate.
s dLemma 2.5. Let a;b2H (R ) with s>1+d=2, then for k2N, we have
(2.6) k[Δ ;a]@ bk s¡1 d •C kak s d kbk s¡1 dk y sH ( ) H ( ) H ( )
and
0(2.7) k[Δ ;a]@ bk s d •C kak s d kbk s d :k y H ( ) s H ( ) H ( )
Proof : We prove only (2.6), following the notations of lemma 2.3, we have
[Δ ;a]@ b=Δ (a@ b)¡aΔ (@ b)k y k y k y
=Δ (T @ b+T a+R(a;@ b))¡(T Δ (@ b)+T a+R(a;Δ (@ b))):k a y @ b y a k y Δ @ b k yy k y
s¡1 s¡1¡d=2 1Since @ b2H ‰C ‰L , (a) and (b) of lemma 2.3 givey
kT a+R(a;@ b)+T a+R(a;Δ (@ b))k s¡1 d •C kak s d kbk s¡1 d :@ b y Δ @ b k y sy y H ( ) H ( )) H ( ))k
RRRRRRRRRLOCAL EXISTENCE 7
On the other hand, there exists N such that0
X
0 0 0 0Δ (T @ b)¡T Δ (@ b)= (Δ (S (a)Δ @ b)¡S (a)Δ (Δ (@ b)));k a y a k y k k k y k k k y
0jk¡kj•N0
and
0 0 0 0Δ (S (a)Δ @ b)¡S (a)Δ (Δ (@ b))(x)k k k y k k k y
R
dk k
0 0 0=2 f(2 (y¡x))(S (a)(y)¡S (a)(x))Δ @ b(y)dy :k k k y
Hence
0 0 0 0 2kΔ (S (a)Δ @ b)¡S (a)Δ (Δ (@ b))kk k k y k k k y L
¡k
1 0 s•2 ktf(t)k 1krak kΔ @ bk 2 •C kak kbk s¡1:L L k y L N H H0
This completes the proof of the lemma.
3. Proof of the Theorem
Inthissection,wearegoingtoprovethelocalexistenceofsolutionstatedinTheorem
1.1. The proof is based on the study of a linearized problem. We want to construct a
convergent sequence of solutions to the problem and show that the limit is
thesolutiontothenonlinearproblem(1.8)-(1.11)withthepropertystatedinTheorem
1.1.
nUnderthehypothesisoftheorem1.1,westudynowthesequenceoffunctionsfv gn2
defined inductively as follows.
1(3.1) v =v ;0
n+1 n 2 n+1 n(3.2) v ¡((„ ) v ) =f ;ytt y
n+1 n+1(3.3) (v ;v )j =(v ;v );t=0 0 1t
with
n n fi„ (y;t)=(a +v (y;t)) ;0
nfi+2 fivn n 3fi¡1 n 2 n 2 n t n nf (y;t)=¡fi(a +v ) (v ) + („ ) v + v ¡v :0 y y t tny a +v0
¡fiFor 0<T <a b and1 00
¡ ¢1=22 2M =B kv k +kv k ;s s¡10 1 0 1H H
fi fiwith B =2(8=a ) if a •2, and B =2(2a ) if a >2, we define1 0 0 1 0 0
n
0 s 0;1 s¡1X = vjv2C ([0;T ];H (R ))\C ([0;T ];H (R ));s;T 1 + 1 +1
‡ ·1=2
2 2jjjvjjj = kvk +kvk •M ;1 s 1 s¡1X t 0s;T L ([0;T ];H ( )) L ([0;T ];H ( ))1 + 1 +1
o1 fikvk 1 • a ; Supp v‰f(y;t)2R £[0;T ]; y+a t‚b g :L ( £[0;T ]) 0 + 1 0+ 1 0
2
We will prove the following theorem.
RRRN8 XU AND YANG
¡fi nTheorem 3.1. (a) For any s > 3=2;0 < " < b ;0 < T • a (b ¡"), if v 2 X ,0 1 0 s;T0 1
then the Cauchy problem (3.2)-(3.3) has a solution
n+1 0 s 0;1 s¡1v 2C ([0;T ];H (R ))\C ([0;T ];H (R ));1 + 1 +
with
n+1 fiSupp v ‰f(y;t)2[";+1[£[0;T ]; y+a t‚b g:1 00
¡fi n(b) For any s > 3=2, there exists 0 < T < a b , such that if v 2 X , then1 0 s;T0 1
n+1the solution v of Cauchy problem (3.2)-(3.3) belongs to X , that means that thes;T1
nsequence fv g is well-defined and uniformly bounded in X .s;T1
n(c) There exists 0 < T • T such that the sequence fv g is a Cauchy sequence in2 1
X .s¡1;T2
¡fi nProof. First for part (a), since 0 <T •a (b ¡"), if v 2X , then1 0 s;T0 1
n fiSupp v ‰f(y;t)2[";+1[£[0;T ]; y+a t‚b g:1 00
We have that
n 0 s 0;1 s¡1„ 2C ([0;T ];H (R))\C ([0;T ];H (R));1 1
and
n 0 s¡1 0;1 s¡2f 2C ([0;T ];H (R))\C ([0;T ];H (R)):1 1
Thus, the existence theorem for linear Cauchy problem gives the existence of solution
to (3.2) and (3.3), cf. [9]
n+1 0 s 0;1 s¡1v 2C ([0;T ];H (R))\C ([0;T ];H (R)):1 1
n n fi nMoreover for v 2 X , we have „ (y;t) = a ;f (y;t) = 0;v = v = 0 for alls;T 0 11 0
fi n+1(y;t) 2 R £ [0;T ]; y + a t • b , so that in this domain, v is the solution of+ 1 00
problem
n+1 2fi n+1v ¡a v =0; (v;v )j =(0;0):t t=0tt 0 yy
n+1Then v =0 in this domain and this gives part (a).
1 1We now turn to part (b). For 0 <"<b , take ´2C (R);´(y)= if y‚";´(y)=0 y
2 fiif y•"=2. We suppose always 0<T •a (b ¡"). For v¯2X , we set1 0 s;T0 1"
fi„¯(y;t)=(a +v¯(y;t)) ;0
2fi+2 fiv¯3fi¡1 2 2 t¯f(y;t)=¡fi(a +v¯) (v¯) + („¯) v¯ + ¡v¯:0 y ty
y a +v¯0
1Remark that ´(y)v¯ = v¯ , since v¯ (y;t)=0 if y•". Then by using Theorem 2.4, wey y yy
have
0 s 0;1 s¡1„¯2C ([0;T ];H (R))\C ([0;T ];H (R));1 1
0 s¡1 0;1 s¡2¯f 2C ([0;T ];H (R))\C ([0;T ];H (R)):1 1
And
B2 [s]+1 [s]¯
1 s¡1 1 s(3.4) kfk • M ; k„¯k •B M ;L ([0;T ];H ( )) 2L ([0;T ];H ( ))1 0 1 0"
RRLOCAL EXISTENCE 9
with constant B depends only on fi;a ;b ands. We consider now the following linear2 0 0
problems
2 ¯(3.5) v ¡((„¯) v ) =f;tt y y
(3.6) (v;v )j =(v ;v );t t=0 0 1
In fact, part (b) is equivalent to the following claim.
Claim: Suppose that
1
Suppv ; Suppv ‰[b ;+1[; kv k 1 • a ;0 1 0 0 L ( ) 0+
4
3and s> . There exists T >0 depending on a ;b ;s;M , such that, for any v¯2X ,1 0 0 0 s;T12
the solution v of problem (3.5)-(3.6) is also in X .s;T1
For the above claim, we only need to prove the following estimate for the solution v
of problem (3.5)-(3.6).
2 2 2(3.7) kvk +kvk •M :1 s 1 s¡1tL ([0;T ];H ( )) L ([0;T ];H ( )) 01 1
1By using Sobolev embedding theorem, Lipschitz estimate and L boundedness of v ,0
we get immediately, for T •a =(4M C ),1 0 0 s
1
kvk 1 • a :L ( £[0;T ]) 01 2
To apply the Lemma 2.1, we need the following estimate,
2 2 2 ¡2k(s¡1)(3.8) kΔ vk +kΔ v k •c 2 ;1 2 1 2k t k yL ([0;T ];L ( )) L ([0;T ];L ( )) k1 1
P
2 2with c •M for k2N.k 0
1 s¡2Sincev 2L ([0;T ];H (R)),byapplyingΔ totheequation(3.5)andintegratingtt 1 k
its product with Δ v over (y;t) inR£[0;t], we have,k t
Z Z Z
1 1 12 2 2 2jΔ vj (y;t)dy+ „¯ jΔ v j (y;t)dy = jΔ v j (y)dyk t k y k 1
2 2 2
Z Z Zt
1 2 2 ¯+ „¯ jΔ (v ) j (y)dy+ Δ (f)Δ (v )dydtk 0 y k k t
2 0
Z Z Z Zt t
2 2+ „¯„¯jΔ (v )j dydt¡ [Δ ;„¯ ]v Δ (v )dydt:t k y k y k ty
0 0
Using Cauchy-Schwarz inequality, we have
12 2 2 2 2kΔ vk (t)+kΔ v k (t)• B (kΔ v k +kΔ (v ) k )2 2 2 2k t k y k 1 k 0 yL ( ) L ( ) 1 L ( ) L ( )4
2 2 2 2 2 k 2 2¯ e+T B kΔ (f)k +T B 2 kΔ ([Δ ;„¯ ]v )k1 2 1 2k k k y1 1 L ([0;T ];L ( )) 1 1 L ([0;T ];L ( ))1 1
1 12 2 2
1+ kΔ (v)k 1 2 + T B k„¯„¯k kΔ v k 1 2 ;k t 1 t L ( £[0;T ]) k yL ([0;T ];L ( )) 1 1 L ([0;T ];L ( ))1 12 4
P
e e0where Δ = Δ , and Δ –Δ =Δ . We have0k k k k kjk¡kj•N1
2fi¡1 2fi¡1
1 1k„¯„¯k •fi(2a ) kvk •fi(2a ) C kvk 1 s¡1 ;t L ( £[0;T ]) 0 t L ( £[0;T ]) 0 s t L ([0;T ];H ( ))1 1 1
RRRRRRRRRRRRRRRRRRRRRRRRR10 XU AND YANG
where s¡1>1=2. By choosing 0<T small enough satisfying1
2 2fi¡1 2 2fi¡1T B fi(2a ) C kvk 1 s¡1 •T B fi(2a ) C M •2;1 0 s t L ([0;T ];H ( )) 1 0 s 01 1 1
we have
1
2 2 2 2 2kΔ vk 1 2 +kΔ v k 1 2 • B (kΔ v k 2 +kΔ (v ) k 2 )k t k y k 1 k 0 y1L ([0;T ];L ( )) L ([0;T ];L ( )) L ( ) L ( )1 1 2
2 2 2 2 2 2k 2 2¯ e+2T B kΔ (f)k +2T B 2 kΔ ([Δ ;„¯ ]v )k :1 2 1 2k k k y1 1 L ([0;T ];L ( )) 1 1 L ([0;T ];L ( ))1 1
Hence, (2.7) and (3.4) yield
¡k(s¡1)¯ ¯
1 2 1 s¡1kΔ (f)k • c 2 kfkk kL ([0;T ];L ( )) L ([0;T ];H ( ))1 1
B2 [s]+1 ¡k(s¡1)• M c 2 ;k0
"
2 ¡ks 2e 1 2 1 s 1 skΔ ([Δ ;„¯ ]v )k • c 2 k„¯ k kvkk k y L ([0;T ];L ( )) k L ([0;T ];H ( )) L ([0;T ];H ( ))1 1 1
[s] ¡ks• B M c 2 kvk 1 s ;2 k L ([0;T ];H ( ))0 1
pP [s]2with c • 1. By choosing 0 < T B B M • 2=4 in the above estimate , we1 1 2k 0
complete the proof of the claim and then obtain the part (b).
nFinally, we want to prove part (c) of Theorem 3.1. Let fv g be a sequence of
functions defined by (3.1)-(3.3), we prove that there exists 0 < T • T such that it2 1
0 s¡1 0;1 s¡2is a Cauchy sequence in C ([0;T ];H (R))\C ([0;T ];H (R)). In fact we will2 2
prove the following estimate, for any n2N,
n+1 n 2 n+1 n 2 ¡n 2(3.9) kv ¡v k +kv ¡v k •2 M :1 s¡1 1 s¡2L ([0;T ];H ( )) t t L ([0;T ];H ( )) 02 2
n+1 n+1 nSet u =v ¡v ;n2N, we have
n+1 n 2 n+1 n n¡1 n 2 n¡1 2 nu ¡((„ ) u ) =(f ¡f )¡(((„ ) ¡(„ ) )u )y ytt y y
n+1 n+1(u ;u )j =(0;0):t=0t
n+1 n n¡1where v ;v ;v 2X , ands;T1
n 2 n¡1 2 n n¡1 n(„ ) ¡(„ ) =b (v ;v )u ;1
n n¡1 n n¡1 n n¡1 nf ¡f =b (v ;v ;@v ;@v )u2
n n¡1 n n¡1 n n n¡1 n n¡1 n+b (v ;v ;@v ;@v )u +b (v ;v ;@v ;@v )u ;3 4t y
with
kb k 1 s¡1 •A(";M ); j =1;¢¢¢ ;4:j L ([0;T ];H ( )) 01
For t2[0;T];0<T •T ,1
fl flZ Z
tfl fl
n n¡1 n+1 ¡k(s¡2) n+1fl fl 1 2Δ (f ¡f )Δ u dydt •TA(";M )c 2 kΔ u kk k 0 k k L ([0;T];L ( ))t tfl fl
0
¡ ¢
n n
1 s¡1 1 s¡2£ ku k +ku k :L ([0;T];H ( )) L ([0;T];H ( ))t
RRRRRRRRRRRRRRRRRRRR