Global Well Posedness for Certain Density Dependent Modified Leray Models

Journal of Inequalities and Applications Volume 2011, Article ID 946208,7pages doi:10.1155/2011/946208

Research Article

Global Well-Posedness for Certain

Density-Dependent Modified-Leray-

α

Models

Wenying Chen

_{and Jishan Fan}

1_{College of Mathematics and Computer Science, Chongqing Three Gorges University,}

Wanzhou, Chongqing 404000, China

2_{Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China} 3_{Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan}

Correspondence should be addressed to Wenying Chen,[email protected]

Received 3 October 2010; Accepted 16 January 2011

Academic Editor: R. N. Mohapatra

Copyrightq2011 W. Chen and J. Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global well-posedness result is established for both a 3D density-dependent modified-Leray-α

model and a 3D density-dependent modified-Leray-α-MHD model.

1. Introduction

A density-dependent Leray-αmodel can be written as

ρtdiv

ρu0,

ρvtρu· ∇v∇π−Δv0,

v1−α2Δu, in0,∞×Ω,

divvdivu0, in0,∞×Ω,

vu0 on 0,∞×∂Ω,

ρ, ρv_t0ρ0, ρ0v0

inΩ⊆R3_,

1.1

whereρis the fluid density,vis the fluid velocity field,uis the “filtered” fluid velocity, andπ

of the filter, and for simplicity, we will takeα≡1.Ω⊆R3_{is a bounded domain with smooth} boundary∂Ω.

Whenρ≡1, the above system reduces to the well-known Leray-αmodel and has been studied in1,2 . Whenα → 0, the above system reduces to the classical density-dependent Navier-Stokes equation, which has received many studies3–6 . Specifically, it is proved in3,

4 that the density-dependent Navier-Stokes equations has a unique locally smooth solution

ρ, vif the following two hypothesesH1andH2are satisfied:

H1ρ0∈W1,qfor someq∈3,6 , v0 ∈H₀1∩H2, and divv00 inR3,

H2∃πandg ∈L2_{such that−Δv}

0∇π ρ1/2₀ ginΩ.

One of the aims of this paper is to prove a global well-posedness result for the density-dependent Leray-αmodel1.1.

Theorem 1.1. Let (H1) and (H2) be satisfied. Then the problem1.1has a unique smooth solution

ρ, π, vsatisfying

ρ∈L∞0, T;W1,q, ρt∈L∞0, T;Lq,

π∈L∞0, T;H1∩L20, T;W1,6,

v∈L∞0, T;H2∩L20, T;W2,6,

ρvt∈L∞

0, T;L2_{, v} t∈L2

0, T;H1

0

,

1.2

for anyT >0.

Next, we consider the following density-dependent modified-Leray-α-MHD model:

ρtdiv

ρu0, 1.3

ρvtρu· ∇v∇π−Δv Bs· ∇B, 1.4

∂tBsu· ∇B−Bs· ∇v ΔB, 1.5

v1−α2Δu, B1−α2_MΔBs, 1.6

divvdivudivBdivBs0, in0,∞×Ω, 1.7

vu0, B·nBs·ncurlB×ncurlBs×n0, on∂Ω, 1.8

ρ, v, Bs_t0

ρ0, v0, Bs0

inΩ⊆R3, 1.9

where B and Bs represent the unknown magnetic field and the “filtered” magnetic field, respectively. αM > 0 is the lengthscale parameter representing the width of the filter and we will takeαM 1 for simplicity.nis the unit outward vector to ∂Ω. Whenα → 0 and

ρ 1 andαM 0, the above system has been studied in10 recently, and also modified models were analyzed in11 . In this paper, we will prove the following theorem.

Theorem 1.2. Let0 < m≤ ρ0 ≤ M < ∞, ρ0 ∈W1,qwith q∈ 3,6 , v0 ∈ H01∩H2, B0 ∈H3, anddivv0divu0divB0divBs00inΩ. Then the problem1.3–1.9has a unique smooth solutionρ, π, v, B, Bssatisfying

0< m≤ρ≤M <∞, ρ∈L∞0, T;W1,q, ρt∈L∞0, T;Lq,

π∈L∞0, T;H1∩L20, T;W1,6,

v∈L∞0, T;H2∩L20, T;W2,6, vt∈L∞

0, T;L2∩L20, T;H₀1,

B∈L∞0, T;H3, ∂tBs∈L∞

0, T;H1, ∂tB∈L2

0, T;H1,

1.10

for anyT >0.

For other related models, we refer to12–16 .

Since the proof ofTheorem 1.1is similar to and simpler than that ofTheorem 1.2, we only proveTheorem 1.2for concision.

2. Proof of

Theorem 1.2

By similar argument as that in3,4 , it is easy to prove that there areT0 > 0 and a unique smooth solutionρ, v, B, Bsto the problem1.3–1.9in0, T0 , and we only need to establish some a priori estimates for any time. Therefore, in the following estimates, we assume that the solutionρ, v, B, Bsis sufficiently smooth.

First, it follows from1.3,1.7, and the maximum principle that

0< m≤ρx, t≤M <∞. 2.1

Testing1.4and1.5byvandB, respectively, using1.3,1.6, and1.7, summing up them, we see that

1 2

d dt ρv

2_|B

s|2|∇Bs|2dx |∇v|2|∇B|2dx0. 2.2

Hence

u_L∞_0,T;H2u_L2_0,T;H3≤C, 2.3

v_L∞_0,T;L2v_L2_0,T;H1≤C, 2.4

BsL∞_0,T;H1B_sL2_0,T;H3≤C, 2.5

Taking∂ito1.3, multiplying it by|∂iρ|q−2∂iρ, summing overi, using1.7and2.3, we have

d dt

∇ρqdx≤C∇u_L∞∇ρq_Lq ≤Cu_H3∇ρq_Lq, 2.7

which yields

_ρ

L∞0,T;W1,q≤C. 2.8

Using1.3,2.3and2.8, we find that _ρt

L∞0,T;Lq ≤u∇ρ_L∞_0,T;Lq≤ u_L∞∇ρ_L∞_0,T;Lq≤C∇ρ_L∞_0,T;Lq≤C. 2.9

Multiplying1.5by−ΔB, using1.6,1.7,2.3, and2.4, we obtain

1 2

d

dt |∇Bs|

2_|ΔB

s|2dx |ΔB|2dx

u· ∇B−Bs· ∇v ΔB dx

≤u_L∞∇B_L2BsL∞∇vL2ΔB_L2

≤C∇B_L2B_sH2∇v_L2ΔB_L2

≤CB1/2_L2 ΔB

1/2

L2 BsH2∇v_L2

≤ 1 2ΔB

2

L2CB2_L2C∇v2_L2Bs2_H2,

2.10

which yields

BsL∞0,T;H2BsL2_0,T;H4≤C, 2.11

B_L∞_0,T;L2B_L2_0,T;H2≤C. 2.12

Multiplying1.4byvt, using1.3,2.11,2.12,2.1,2.3, and2.4, we have

1 2

d dt |∇v|

2_{dx ρv}2

tdx Bs· ∇B·vtdx− ρu· ∇v·vtdx

≤ BsL∞∇B_L2vtL2ρ_L∞· uL∞· ∇v_L2·ρvt_L2

≤C∇B_L2·ρvt_L2C∇vL2ρvt_L2

≤ 1

2ρvt 2

L2C∇B

2

L2C∇v2_L2,

which implies

v_L∞_0,T;H1u_L∞_0,T;H3≤C, 2.14

vtL2_0,T;L2≤C. 2.15

It follows from1.4,2.14,2.15,2.11,2.12, and theH2-theory for Stokes system that17

v_L2_0,T;H2u_L2_0,T;H4 ≤C. 2.16

Similarly, it follows from1.5,2.11,2.12, and2.16that

∂tBsL2_0,T;L2≤C. 2.17

Taking∂t to1.5, multiplying it by ∂tB, using1.7,1.8,2.12,2.11,2.14, and

2.15, we get

1 2

d

dt |∂tBs|

2_|∇∂

tBs|2dx |∇Bt|2dx

− ut· ∇B·Btdx ∂tBs· ∇v·Btdx Bs· ∇vt·Btdx

ut∇Bt·Bdx ∂tBs· ∇v·Btdx− Bs· ∇Bt·vtdx

≤ utL∞∇BtL2B_L2∂tBsL3· ∇v_L2· BtL6BsL∞∇BtL2vtL2

≤CvtL2∇BtL2C∂tBsH1∇BtL2

≤ 1 2∇Bt

2

L2Cvt2L2C∂tBs2H1,

2.18

which implies

∂tBsL∞_0,T;H1∂tBsL2_0,T;H3≤C, 2.19

BtL2_0,T;H1≤C. 2.20

Due to1.5,2.3,2.11,2.12,2.14,2.19,2.16, and theH2_{-theory of the elliptic} equations, we have

B_L∞_0,T;H2B_L2_0,T;H3≤C, 2.21

Taking∂tto1.4, we see that

ρvttρu· ∇vt∇πt−Δvt∂tBs· ∇BBs· ∇∂tB−ρtvt−

ρtuρut

· ∇v. 2.23

Multiplying the above equation by vt, using 1.3, 2.19, 2.21, 2.22, 2.9, and

2.14, we deduce that

1 2

d dt ρv

2

tdx |∇vt|2dx

≤ ∂tBsL6· ∇B_L2· vtL3

BsL∞· ∇∂tBL2· vtL2ρt_Lq· vtL2q/q−2· vtL2

ρt_Lq· u_L∞· ∇v_L2· vtL2q/q−2ρ_L∞utL∞· ∇v_L2· vtL2

≤CvtL3C∇∂tBL2vtL2CvtL2q/q−2vtL2CvtL2q/q−2Cvt2L2

≤ 1 2∇vt

2

L2Cvt2L2C∇∂tB2L2C,

2.24

which gives

vtL∞_0,T;L2vtL2_0,T;H1

0≤C. 2.25

Combining1.4,2.21,2.22,2.25,2.14, and the regularity theory of the Stokes system17 , we obtain

v_L∞_0,T;H2v_L2_0,T;W2,6≤C,

π_L∞_0,T;H1π_L2_0,T;W1,6≤C,

u_L∞_0,T;H4u_L2_0,T;W4,6≤C.

2.26

Similarly, one can prove that

B_L∞_0,T;H3≤C. 2.27

This completes the proof.

Acknowledgment

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