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volume 6, issue 4, article 121, 2005.

Received 22 February, 2005;

accepted 01 September, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

NORMIC INEQUALITY OF TWO-DIMENSIONAL VISCOSITY OPERATOR

YUEHUI CHEN

Department of Mathematics Zhangzhou Teachers College Zhangzhou 363000, P.R. China.

EMail:yuehuich@21cn.com

c

2000Victoria University ISSN (electronic): 1443-5756 051-05

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Normic Inequality of Two-Dimensional Viscosity

Operator Yuehui Chen

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J. Ineq. Pure and Appl. Math. 6(4) Art. 121, 2005

Abstract

A relation between the coefficients of Legendre expansions of two-dimensional function and those for the derived function is given. With this relation the normic inequality of two-dimensional viscosity operator is obtained.

2000 Mathematics Subject Classification:65N35.

Key words: Legendre expansion, Two-dimension, Viscosity operator.

The author wants to express his deep gratitude to Prof. Benyu Guo for research inspi- ration and is greatly indebted to the referee for improving the presentation. The work is supported by the Science Foundation of Zhangzhou Teachers College (JB04302).

1. Introduction

Spectral methods employ various orthogonal systems of infinitely differentiable functions to represent an approximate projection of the exact solution sought.

The resulting high accuracy of spectral algorithms was a major motivation be- hind their rapid development in the past three decades, e.g., see Gottlieb and Orszag [1] and Guo [2] .

For nonperiodic problems, it is natural to use Legendre spectral methods or Legendre pseudospectral methods. More attention has been paid to these two methods recently due to the appearance of the Fast Legendre Transformation. In studying the spectral methods for nonlinear conservation laws, we have to face those equations whose solutions may develop spontaneous jump discontinuities, i.e., shock waves. To overcome these difficulties, the spectral viscosity (SV) method was introduced by Tadmor [3] . Maday, Ould Kaber and Tadmor [4]

firstly considered the nonperiodic Legendre pseudospectral viscosity method for an initial-boundary value problem, and Ma [5] , Guo, Ma and Tadmor [6]

recently developed the nonperiodic Chebyshev-Legendre approximation. So far, however, few works have been done in multiple dimensions. This paper will study the relation between the coefficients of Legendre expansions of the two- dimensional function and those for the derived function, and gives the normic inequality of the two-dimensional SV operator, which plays important roles in the SV method.

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2. Notations and Lemmas

Letx= (x₁, x₂),Λ = (−1,1)².We define the spaceL^p(Λ)and its normk · k_L^p in the usual way. Ifp= 2, we denote the norm of spaceL²(Λ)byk · k,that is

kvk= Z

Λ

|v(x)|²dx ¹₂

.

LetN be the set of all non-negative integers andP_N be the set of all algebraic polynomials of degree at mostN in all variables. Letl = (l₁, l₂) ∈ N²,|l| = max{l1, l2}, the Legendre polynomial of degreel is Ll(x) = Ll1(x1)Ll2(x2).

The Legndre transformation of a functionv ∈L²(Λ)is Sv(x) =

∞

X

|l|=0

ˆ v_lL_l(x),

with the Legendre coefficients

ˆ v_l =

l₁+ 1

2 l₂+1 2

Z

Λ

v(x)L_l(x)dx, |l|= 0,1, . . . . Lemma 2.1 ([2]). For anyφ ∈ P_N,2≤p≤ ∞and|k|=m,we have

k∂_x^kφkL^p ≤cN^2mkφkL^p.

Here,cis a generic positive constant independent of any function andN.

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A viscosity operatorQis defined by

(2.1) Qv(x) :=

N

X

|l|=0

ˆ

qlˆvlLl(x), v =

∞

X

|l|=0

ˆ vlLl(x).

Here,qˆ_lare the so-called viscosity coefficients,





 ˆ

q_l = 0 for |l| ≤m ˆ

q_l ≥1−_l2^m²

1+l²₂ for m <|l| ≤N.

Observe that theQoperator is activated by only the high mode numbers,≥ m.

In particular, if we letm−→ ∞,theQoperator is spectrally small (in the sense thatkQvk_H^−s ≤cm^−skvk).

LetRdenote the corresponding low modes filter

(2.2) Rv(x) :=

N

X

|l|=0

ˆ

r_lvˆ_lL_l(x), hererˆ_l= 1−qˆ_l.

Clearly,





 ˆ

r_l= 1 for |l| ≤m ˆ

r_l≤ _l2^m²

1+l²₂ for m <|l| ≤N.

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3. Main Results

Firstly , we consider the relation between the coefficients of Legendre expansions ofv(x)and those for∂xv(x).

Theorem 3.1. For anyv(x)∈H¹(Λ),letvˆ_k,vˆ⁽¹⁾_k be the coefficients of Legendre expansions ofv(x), ∂_xv(x), |k|= 0,1,2, . . . ,

ˆ

v_k⁽¹⁾ = ˆv⁽¹⁾_(k

1,k2) = (2k1+ 1)

∞

X

p=k1+1 k1+podd

ˆ

v_(p,k₂₎+ (2k2+ 1)

∞

X

q=k2+1 k2+qodd

ˆ v_(k₁_,q).

Proof. By the property of the one-dimensional Legendre polynomial:

(2k+ 1)L_k(x) = L⁰_k+1(x)−L⁰_k−1(x), k ≥1, we have

L⁰_k(x) =

k−1

X

l=0 k+lodd

(2l+ 1)L_l(x), k= 0,1,2, . . . .

Then, forx= (x1, x2),

∂_xL_k(x) = L⁰_k

1(x₁)L_k₂(x₂) +L_k₁(x₁)L⁰_k

2(x₂)

=

k1−1

X

l1=0 k1+l1 odd

(2l₁+ 1)L_l₁(x₁)L_k₂(x₂) +

k2−1

X

l2=0 k2+l2 odd

(2l₂+ 1)L_l₂(x₂)L_k₁(x₁);

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∂_xv =

∞

X

|k|=0

ˆ

v_k∂_xL_k(x)

=

∞

X

k1=0

∞

X

k2=0

ˆ v_(k₁_,k₂₎







k1−1

X

l1=0 k1+l1 odd

(2l₁+ 1)L_l₁(x₁)L_k₂(x₂)

+

k2−1

X

l2=0 k2+l2 odd

(2l₂+ 1)L_l₂(x₂)L_k₁(x₁)





.

On the other hand,

∂_xv =

∞

X

|k|=0

ˆ

v⁽¹⁾_k L_k(x) =

∞

X

k1=0

∞

X

k2=0

ˆ v⁽¹⁾_(k

1,k2)L_k₁(x₁)L_k₂(x₂).

We can obtain

ˆ v⁽¹⁾_(k

1,k2) = (2k1+ 1)

∞

X

p=k1+1 k1+podd

ˆ

v_(p,k₂₎+ (2k2+ 1)

∞

X

q=k2+1 k2+qodd

ˆ v_(k₁_,q).

Specially, for anyv ∈ P_N,we have

(3.1) vˆ⁽¹⁾_(k

1,k2) = (2k₁+ 1)

N

X

p=k1+1 k1+podd

ˆ

v_(p,k₂₎+ (2k₂+ 1)

N

X

q=k2+1 k2+qodd

ˆ v_(k₁_,q).

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LetJ_k,N ={j

k+ 1≤j ≤N, k+j odd},then ˆ

v⁽¹⁾_(k

1,k2) = (2k₁+ 1) X

p∈J_k

1,N

ˆ

v_(p,k₂₎+ (2k₂+ 1) X

q∈J_k

2,N

ˆ

v_(k₁_,q) ∀v ∈ P_N

Theorem 3.2. Consider the SV operatorQ =Q_m,(2.1) with the parametriza- tion and operatorR,(2.2). For anyφ ∈ P_N , the following inequalities hold:

k∂_x(Rφ)k² ≤cm⁴lnNkφk²

k∂xφk² ≤2k∂x(Qφ)k² +cm⁴lnNkφk² k∂_x(Qφ)k² ≤2k∂_xφk²+cm⁴lnNkφk² Proof. We decompose∂_x(Rφ(x)) =A₁(x) +A₂(x), where

A₁(x) :=∂_x





m

X

|k|=0

ˆ

r_kφˆ_kL_k(x)



, A₂(x) :=∂_x





N

X

|k|=m+1

ˆ

r_kφˆ_kL_k(x)



.

By Lemma ,k∂_xφk ≤cN²kφk, ∀φ(x)∈ P_N,and hencekA₁(x)k² ≤cm⁴kφk². Further letJ_k,N⁽⁰⁾ ={j

j ∈J_k,N, j > m}, J_k,N^(d) ={j

j ∈J_k,N,max{j, k_d}>

m},here,m∈ N, d= 1,2.Then

kA₂(x)k² ≤

N

X

|k|=0





(2k₁+ 1) X

p∈J_k⁽²⁾

1,N

ˆ

r_(p,k₂₎φˆ_(p,k₂₎

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+(2k₂+ 1) X

q∈J_k⁽¹⁾

2,N

ˆ

r_(k₁_,q)φˆ_(k₁_,q)







2

kL_kk²

≤8 (A_2,1+A_2,2), in which

A_2,1

=

N

X

k1=0 N

X

k2=0

(2k₁+ 1)²





 X

p∈J_k⁽²⁾

1,N

ˆ







2

1

(2k₁+ 1)(2k₂+ 1)

=

N

X

k1=0 N

X

k2=0

2k₁+ 1 2k₂+ 1





 X

p∈J_k⁽²⁾

1,N

ˆ







2

≤

N

X

k1=0 N

X

k2=0

2k₁+ 1 2k2+ 1





 X

p∈J_k⁽²⁾

1,N

|ˆr_(p,k₂₎|²kL_(p,k₂₎k⁻²











 X

p∈J_k⁽²⁾

1,N

|φˆ_(p,k₂₎|²kL_(p,k₂₎k²







≤

N

X

k1=0 N

X

k2=0

2k₁+ 1 2k₂+ 1





 X

p∈J_k⁽²⁾

1,N

m⁴(2p+ 1)(2k₂ + 1) 4(p²+k²₂)²

X

p∈J_k⁽²⁾

1,N

|φˆ_(p,k₂₎|²kL_(p,k₂₎k²







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≤c₁m⁴

N

X

k2=0







N

X

k1=0

(2k₁+ 1)





 X

p∈J_k⁽²⁾

1,N

2p+ 1 (p²+k₂²)²







N

X

p=0

|φˆ_(p,k₂₎|²kL_(p,k₂₎k²

!







≤c₂m⁴

N

X

k2=0 N

X

p=0

|φˆ_(p,k₂₎|²kL_(p,k₂₎k²

! _N X

k1=0

(2k₁+ 1) X

p∈J_k⁽⁰⁾

1,N

p⁻³

≤2c₂m⁴

N

X

k2=0 N

X

p=0

|φˆ_(p,k₂₎|²kL_(p,k₂₎k² m⁻²

m

X

k1=0

(2k₁+ 1) +

N

X

k1=m+1

(2k₁+ 1)k₁⁻²

!

≤c₃m⁴lnN

N

X

k2=0 N

X

p=0

|φˆ_(p,k₂₎|²kL_(p,k₂₎k²

≤c₃m⁴lnNkφk²; Similarly,

A_2,2 =

N

X

k1=0 N

X

k2=0

2k₂+ 1 2k₁+ 1





 X

q∈J_k⁽¹⁾

2,N

ˆ

r_(k₁_,q)φˆ_(k₁_,q)







2

≤c₄m⁴lnNkφk².

Thus

kA₂(x)k² ≤c₅m⁴lnNkφk².

k∂_x(Rφ)k² ≤2kA₁(x)k² + 2kA₂(x)k² ≤cm⁴lnNkφk². Since∂_xφ≡∂_x(Qφ) +∂_x(Rφ),the desired estimates follow.

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Remark 1. The theorem shows the equivalence of theH¹norm before and after application of the SV operator , Q = Qm for moderate size ofm_N N^1/4. This holds despite the fact that form =m_N ∼cN^β −→ ∞, 0 <4β < 1,the corresponding SV operatorQ_m is spectrally small.

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References

[1] D. GOTTLIEB ANDS.A. ORSZAG, Numerical analysis of spectral meth- ods:Theory and applications, SIAM, Philadelphia, 1977.

[2] B.-Y. GUO, Spectral Methods and their Applications, Singapore: World Scientific, 1998.

[3] E. TADMOR. Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), 30–44.

[4] Y. MADAY, S.M. OULD KABER, AND E. TADMOR, Legendre pseu- dospectral viscosity method for nonlinear conservation laws, SIAM J. Nu- mer. Anal., 30 (1993), 321–342.

[5] H.-P. MA, Chebychev-Legendre pseudospectral viscosity method for non- linear conservation laws, SIAM J. Numer. Anal., 35 (1998), 869–892.

[6] B.-Y. GUO, H.-P. MA AND E. TADMOR, Spectral vanishing viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 39 (2001), 1254–1268.