Navier-Stokes Equations with Potentials

Volume 2007, Article ID 79406,30pages doi:10.1155/2007/79406

Research Article

Navier-Stokes Equations with Potentials

Received 11 March 2007; Accepted 8 May 2007 Recommended by Viorel Barbu

We study Navier-Stokes equations perturbed with a maximal monotone operator, in a bounded domain, in 2D and 3D. Using the theory of nonlinear semigroups, we prove existence results for strong and weak solutions. Examples are also provided.

Copyright © 2007 Adriana-Ioana Lefter. 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.

1. Introduction

Let T >0 and let Ω⊂Rⁿ,n=2, 3, be an open and bounded domain, with a smooth boundary∂Ω(of classC², e.g.). Consider the perturbed Navier-Stokes equations

∂y

∂t ⁻νΔy+ (y· ∇)y+Φ(y) +∇pg, inQ=Ω×(0,T), divy=0, inQ,

y=0, onΣ=∂Ω×(0,T), y(·, 0)=y0, inΩ,

(1.1)

wherey=(y1,y2,. . .,y_n) is the velocity field, pis the scalar pressure. The density of ex- ternal forces isg=(g1,g2,. . .,gn), the constantν>0 is the kinematic viscosity coefficient, and the perturbationΦis a maximal monotone operator. Such a nonlinear termΦarises usually as a feedback nonlinear controller.

In this section, we describe the functional framework and we rewrite the Navier-Stokes equations in an abstract form. The main existence and uniqueness results for strong so- lutions are stated inSection 2. The first of these theorems is proved inSection 3and the others inSection 4.Section 5is concerned with weak solutions. The last section is devoted to examples.

We will use the standard spaces (see, e.g., [1–3]) H=

y∈

L²(Ω)ⁿ; divy=0 inΩ,y·n_∂Ω=0 on∂Ω, V=

y∈

H₀¹(Ω)ⁿ; divy=0 inΩ. (1.2) H is a real Hilbert space endowed withL²-norm| · | andV is a real Hilbert space endowed withH0¹-norm · = |∇ · |. Moreover, denoting byV the dual space ofV and consideringH identified with its own dual, we haveV ⊂H⊂Valgebraically and topologically with compact injections.

Here, (·,·) denotes the scalar product ofHand the pairing betweenVand its dualV. The norm ofVis denoted by · V.

LetA∈L(V,V) (the space of linear continuous operators fromV inV), (Ay,z)= _n

i=1

Ω∇yi· ∇zidx, for ally,z∈V.

We have (Ay,y)= y², for all y∈V. We setD(A)= {y∈V; Ay∈H}and denote again byAthe restriction ofAtoH.

Letb:V×V×V→Rthe trilinear continuous functional defined by b(y,z,w)=

n

i,j=1 Ωy_i∂z_j

∂x_iw_jdx, ∀y,z,w∈V. (1.3) The functionalbsatisfies (see, e.g., [1–3])

b(y,w,w)=0, b(y,z,w)= −b(y,w,z), ∀y,z,w∈V, (1.4) b(y,z,w)≤C|y|^1/2y^1/2z^1/2|Az|^1/2|w|, ∀y,w∈V,z∈D(A) (n=2), (1.5) b(y,z,w)≤C|y|^1/2y^1/2|z|^1/2z^1/2w, ∀y,z,w∈V(forn=2), (1.6) b(y,z,w)≤Cyz^1/2|Az|^1/2|w|, ∀y,w∈V,z∈D(A) (forn=3), (1.7) b(y,z,w)≤C|y|^1/2y^1/2zw, ∀y,z,w∈V (forn=3), (1.8) b(y,z,w)≤Cyzw, ∀y,z,w∈V(forn=2, 3). (1.9)

LetB:V→Vbe defined by

(B y,w)=b(y,y,w), ∀y,w∈V. (1.10)

In this setting, equations (1.1) may be rewritten d y

dt(t) +νAy(t) +B y(t) +Φ(y(t)) f(t), t∈(0,T) y(0)=y0,

(1.11)

where f =Pg,P: (L²(Ω))ⁿ→His the Leray projection.

SupposeΦsatisfies the following hypotheses:

(h1)Φ=∂ϕ, where ϕ:H→Ris a lower semicontinuous proper convex function (henceΦis a maximal monotone operator inH×H);

(h2) 0∈D(Φ);

(h3) there exist two constantsγ≥0,α∈(0, (1/ν)) such that Au,Φλ(u)≥ −γ1 +|u|²

−αΦλ(u)², ∀λ >0,∀u∈D(A), (1.12) whereΦλ=(1/λ)(I−(I+λΦ)⁻¹) :H→His the Yosida approximation ofΦ.

We consider the classical definition of the maximal monotone operator. We will denote

|Φ(u)| =inf{|z|;z∈Φ(u)}, whereu∈D(Φ).

In the sequel, the symbolwill be used to denote convergence in the weak topology, while the strong convergence will be denoted by→.

2. Main results for strong solutions

Theorem 2.1. LetT >0 and letΩ⊂Rⁿ,n=2, 3 be an open and bounded domain, with a smooth boundary. Assume thatΦ⊂H×H satisfies the hypotheses (h1)–(h3). Let y0∈ D(A)∩D(Φ) and f ∈W^1,1(0,T;H).

Ifn=2, there exists a unique y∈W^1,∞(0,T;H)∩L^∞(0,T;D(A))∩C([0,T];V) such that

d y

dt(t) +νAy(t) +B y(t) +Φy(t)f(t), a.e.t∈(0,T), y(0)=y0.

(2.1)

Moreover,yis right differentiable, (d⁺/dt)yis right continuous, and d⁺

dty(t) +νAy(t) +B y(t) +Φy(t)−f(t)⁰=0, ∀t∈[0,T). (2.2) Ifn=3, the solutionyexists on some interval [0,T0), where

T0=T0

fL²(0,T;H),y0²

≤T. (2.3)

We have denoted byy→(νAy+B y+Φ(y)−f(t))⁰the minimal section of the multi- valued mappingy→(νAy+B y+Φ(y)−f(t)).

If we ask for lower regularity of the initial data, we obtain the following results.

Theorem 2.2 (casen=2). LetT >0 and letΩ⊂R²be an open and bounded domain, with a smooth boundary. Assume that Φ⊂H×H satisfies the hypotheses (h1)–(h3). Let y0∈V∩D(Φ), f ∈L²(0,T;H). Then there exists a unique solution y∈C([0,T];H)∩ L²(0,T;D(A))∩L^∞(0,T;V) withd y/dt∈L²(0,T;H),B y∈L²(0,T;H) for

d y

dt(t) +νAy(t) +B y(t) +Φy(t) f(t), a.e.t∈(0,T), y(0)=y0.

(2.4)

Theorem 2.3 (casen=3). LetT >0 and letΩ⊂R³be an open and bounded domain, with a smooth boundary. Assume that Φ⊂H×H satisfies the hypotheses (h1)–(h3). Let y0∈V∩D(Φ), f ∈L²(0,T;H).

Then there existsT0=T0(fL²(0,T;H),y0²)≤Tsuch that the problem d y

dt(t) +νAy(t) +B y(t) +Φy(t)f(t), a.e.t∈ 0,T0

y(0)=y0

(2.5)

has a unique solution

y∈C0,T0

;H∩L²0,T0;D(A)∩L^∞0,T0;V, d y

dt ^∈L²0,T0;H, B y∈L²0,T0;H.

(2.6)

Remark 2.4. We obtain the same results ifΦsatisfies the following hypotheses:

(H1)Φis a single-valued maximal monotone operator inH×H;

(H2) there exist three constantsγ1,γ2≥0, andα∈(0,ν) such that

Φ(u)≤α|Au|+γ1u+γ2, ∀u∈D(A). (2.7) In the sequel, we use the same symbolCfor various positive constants.

3. Proof ofTheorem 2.1

The proof uses the theory of nonlinear differential equations of accretive type in Ba- nach spaces. In order to obtain a quasi-m-accretive operator in the left-hand side of the Navier-Stokes equation (Proposition 3.1), we have to substitute the nonlinearityBwith a truncationB_N,N∈N^∗. We may then state existence and uniqueness results for the approximate equations (3.2), (3.34) involvingBN,Φ, andBN,Φλ,λ >0 instead ofB,Φ (Propositions3.2,3.3).

We intend to prove that forNlarge enough, the solution of the truncated problem in- volvingBN,Φcoincides with the solution of the initial problem. To this aim, we need to obtain estimates on the solutionyNof problem (3.2). In order to do this, we are obliged to deduce the convenient estimates first on problem (3.34) (the one involvingΦλ) be- cause relation (1.12) does not extend in a suitable way to arbitrary elements ofΦ(y_N(t)).

Passing to the limit withλ→0 in (3.34), we return to the problem inBN,Φand conclude the proof.

3.1. Approximate problems: existence and uniqueness. ForN∈N^∗, define the modi- fied nonlinearityB_N:V→V,

BNy=

⎧⎪

⎪⎨

⎪⎪

⎩

B y ify ≤N, N

y 2

B y ify> N, (3.1)

and consider the approximate equation d yN

dt (t) +νAyN(t) +BNyN(t) +ΦyN(t) f(t), t∈(0,T) yN(0)=y0.

(3.2) Proposition 3.1is one of the main ingredients of the proof.

Proposition 3.1. LetN∈N^∗be fixed. LetΦ⊂H×Hbe a maximal monotone operator with 0∈D(Φ). Assume that there exist two constantsγ≥0,α∈(0, 1/ν) such that relation (1.12) is verified.

Define the operatorΛN:D(ΛN)→H,ΛN=νA+B_N+Φ+α_NI,α_N>0, whereD(ΛN)= {u∈H;∅ =ΛN(u)⊂H}. ThenD(ΛN)=D(A)∩D(Φ) andΛNis a maximal monotone inH×HforαNlarge enough.

Moreover, there exists a constantC_N>0 such that

|Aw| ≤C_N1 +|w|²+νAw+B_Nw+Φλ(w)^23/2, ∀w∈D(A), ∀λ >0, (3.3)

|Aw| ≤CN

1 +|w|²+νAw+BNw+η^23/2, ∀w∈D(A)∩D(Φ), ∀η∈Φ(w).

(3.4) Proof. It has been proved in [4] (see Lemma 5.1, page 292) thatνA+B_NappliesD(A) into H and that forαN large enough, the operatorΓN=νA+BN+αNI withD(ΓN)=D(A) is (maximal) monotone inH×H. ThenD(A)∩D(Φ)⊂D(ΛN) andΛN=ΓN+Φis the sum of two monotone operators, and by consequence it is a monotone. In order to obtain the maximal monotony ofΛN, it is sufficient to prove thatR(I+ΛN)=H.

Let f ∈Handλ >0 a fixed. We approximate the equation

u+νAu+BNu+Φ(u) +αNuf (3.5) by the equation

uλ+νAuλ+BNuλ+Φλ uλ

+αNuλ=f, λ >0, (3.6) that is

uλ+ΓNuλ+Φλ uλ

=f, (3.7)

whereΦλis the Yosida approximation ofΦ. By the properties of the Yosida approxima- tion,Φλis demicontinuous monotone and its sum with the maximal monotone operator ΓNis maximal monotone, which implies the existence of a solutionuλ∈D(A) for (3.6).

The uniqueness follows by monotony arguments.

LetμN=αN+ 1; then (3.6) reads νAu_λ+B_Nu_λ+Φλ

u_λ+μ_Nu_λ=f, λ >0. (3.8) We first multiply (3.8) byuλand infer that

νu_λ²+B_Nu_λ,u_λ+Φλ

u_λ,u_λ+μ_Nu_λ²=

f,u_λ. (3.9)

Butb(uλ,uλ,uλ)=0. Also, the operatorΦλ is monotone, with 0∈D(Φλ)=H, which implies that (Φλ(uλ),uλ)≥(Φλ(0),uλ), and we have

−

Φλ(0),uλ

≤ 1 μN

Φ(0)²+μN

4 uλ². (3.10)

Equation (3.9) yields

νu_λ²+μN

2 u_λ²≤ 1

μ_N^|f|²+ 1

μ_NΦ(0)². (3.11)

Consequently,

uλ²,uλ²≤C1 +|f|²

, ∀λ >0, (3.12)

where the constantC >0 does not depend onλ.

Next, (3.8) is multiplied byAuλ, which gives νAu_λ²+B_Nu_λ,Au_λ+Φλ

u_λ,Au_λ+μ_Nu_λ²=

f,Au_λ. (3.13) ButBNuλ,Auλ≤buλ,uλ,Auλ

≤

⎧⎪

⎨

⎪⎩

Cu_λ^1/2u_λAu_λ^3/2≤ν

4Au_λ²+Cu_λ²u_λ⁴, n=2, Cuλ^3/2Auλ^3/2≤ν

4Auλ²+Cuλ⁶, n=3,

≤ν

4Au_λ²+C1 +|f|²3

,

(3.14) whereC >0 denotes several positive constants (not depending onλ). We used estimates (1.5) in the casen=2, (1.7) in the casen=3, then Young inequality and (3.12).

Recalling also hypothesis (1.12), (3.13) implies that νAu_λ²−ν

4Au_λ²−C1 +|f|²3

−γ1 +|u_λ|²

−αΦλ

u_λ²+μ_Nu_λ²

≤1

ν^|f|²+ν

4Auλ².

(3.15)

Ignoring the termμ_Nu_λ²≥0, by (3.12) the above relation reads ν

2Au_λ²≤αΦλ

u_λ²+C1 +|f|²3

, ∀λ >0, (3.16) whereC >0 denotes several positive constants (not depending onλ).

Finally, we multiply (3.8) byΦλ(u_λ) and obtain νAuλ,Φλ

uλ

+BNuλ,Φλ

uλ

+Φλ

uλ²+μN

uλ,Φλ

uλ

= f,Φλ

uλ

. (3.17)

As shown before,

μ_Nu_λ,Φλ

u_λ≥ −μ_N

2 Φ(0)²+u_λ². (3.18) Also,

B_Nu_λ,Φλ

u_λ≤bu_λ,u_λ,Φλ

u_λ

≤

⎧⎪

⎨

⎪⎩

Cu_λ^1/2u_λAu_λ^1/2Φλ

u_λ), n=2, Cuλ^3/2Auλ^1/2Φλ

uλ

|, n=3,

≤1−να 4 Φλ

uλ²+C1 +|f|²3/2Auλ.

(3.19)

We used again (1.5) in the casen=2, (1.7) in the casen=3, and (3.12). The constantsC do not depend onλ. Together with (1.12), (3.17) implies that

−νγ1 +uλ²

−ναΦλ

uλ²−1−να 4 Φλ

uλ²−C1 +|f|²_3/2Auλ

+Φλ

uλ²−μN

2 Φ(0)²+uλ²

≤ 1

1−να^|f|²+1−να 4 Φλ

uλ², (3.20) and by (3.12),

Φλ

uλ²≤C1 +|f|²3/2Auλ+C1 +|f|²

, ∀λ >0. (3.21) Substituting (3.21) into (3.16), we obtain

ν

2Auλ²≤C1 +|f|²3/2Auλ+C1 +|f|²3

, (3.22)

which implies that

Au_λ≤C1 +|f|²3/2

, Φλ

u_λ²≤C1 +|f|²3

.

(3.23) The constantsCdo not depend onλor|f|.

From the boundedness in H of the sequences (uλ)λ>0, (Φλ(uλ))λ>0, (fλ)λ>0, where f_λ= f −u_λ−Φλ(u_λ)=ΓNu_λ, it follows that on a sequence λ_j→0, we have the weak convergences inH:

u_λj u, Φλj

u_λj f1, fλj= ΓNuλj f2. (3.24) Because (uλj) is bounded inVby (3.12), we get thatuλj→u.

Passing to the weak limit in the equality f−u_λj−Φλj(u_λj)= f_λj, we obtain f =u+ f1+ f2. If we prove that f2=ΓNu, f1∈Φ(u), it will follow that f ∈u+ΓNu+Φ(u), as claimed.

We multiply byuλ−uμthe difference of (3.7) written forλ >0 and the same equation written forμ >0. We find

Φλ uλ

−Φμ uμ

,uλ−uμ

+ΓN+Iuλ−

ΓN+Iuμ,uλ−uμ

=0. (3.25)

SinceΓN+I is the sum of two monotone operators and by consequence monotone, we get (Φλ(uλ)−Φμ(uμ),uλ−uμ)≤0, for allλ,μ >0. Then (u,f1)∈Φand

λ,μlim→0

Φλ

u_λ−Φμ

u_μ,u_λ−u_μ=0 (3.26)

(see [5, Proposition 1.3(iv), page 49]).

Relation (3.25) implies that

λ,μlim→0

ΓN+Iuλ−

ΓN+Iuμ,uλ−uμ

=0. (3.27)

Using alsou_λj→u,ΓNu_λj f2and the fact thatΓN+Iis maximal monotone (ΓNmax- imal monotone), it follows that (u,u+ f2)∈ΓN+I, and thusΓNu= f2 (see [5, Lemma 1.3, page 49]).

From (u,f1)∈ΦandΓNu=f2, we also getu∈D(ΓN)∩D(Φ)=D(A)∩D(Φ). Con- sequently,D(ΛN)=D(A)∩D(Φ).

Let us prove now (3.3) and (3.4).

For the first one, we considerλ >0 fixed,w∈D(A), and letg_λ=νAw+B_Nw+Φλ(w) + μNw. In the same way as we deduced (3.23), we may obtain|Aw| ≤C(1 +|gλ|²)^3/2, hence

|Aw|^2/3≤C1 +gλ²

=CνAw+BNw+Φλ(w) +μNw²+ 1

≤C1 + 2μ²_N|w|²+ 2νAw+B_Nw+Φλ(w)²,

(3.28)

where the constantC >0 does not depend onλ. Thus (3.3) is proved.

In order to prove the second relation, we takew∈D(A)∩D(Φ) andη∈Φ(w). Let g=νAw+B_Nw+η+μ_Nw. For thisg, we may construct as in the first part of the proof a sequence (wλ)λ>0⊂Hsuch that

νAw_λ+B_Nw_λ+Φλ

w_λ+μ_Nw_λ=g, ∀λ >0. (3.29) Moreover,wλ→w,AwλAwbecauseΛNis maximal monotone.

Passing to the limit withλ→0 in (3.23) written for (w_λ), we obtain |Aw| ≤C(1 +

|g|²)^3/2, hence

|Aw_|^2/3_≤C1 +|g_|²₌C1 +νAw+B_Nw+η+μ_Nw²

≤C1 + 2μ²_N|w|²+ 2νAw+B_Nw+η²,

(3.30)

which proves relation (3.4). This concludes the proof ofProposition 3.1.

Proposition 3.2. Let Φ⊂H×H verifying the hypotheses in Proposition 3.1. Let f ∈ W^1,1(0,T;H) andy0∈D(A)∩D(Φ). Then there exists a unique strong solution

y_N∈W^1,∞(0,T;H)∩L^∞0,T;D(A)∩C[0,T];V (3.31) to problem (3.2). Moreover,yNis right differentiable, (d⁺/dt)yNis right continuous, and

d⁺

dtyN(t) +νAyN(t) +BNyN(t) +ΦyN(t)−f(t)⁰=0, ∀t∈[0,T). (3.32) Proof. FromProposition 3.1and [5, Theorems 1.4, 1.6, pages 214–216], it follows that problem (3.2) has a unique solutionyN∈W^1,∞(0,T;H) verifying relation (3.32). In or- der to prove thatyN∈L^∞(0,T;D(A))∩C([0,T];V), letζN(t)∈Φ(yN(t)) such that

d yN

dt (t) +νAy_N(t) +B_Ny_N(t) +ζ_N(t)=f(t). (3.33) We knowf−(d y_N/dt)∈L^∞(0,T;H). Consequently,νAy_N+B_Ny_N+ζ_N∈L^∞(0,T;H).

Applying (3.4) for yN(t) and ζN(t)∈Φ(yN(t)), we get AyN∈L^∞(0,T;H), which im- plies thaty_N∈L^∞(0,T;D(A)). Together with (d y_N/dt)_∈L^∞(0,T;H), we infer thaty_N∈

C([0,T];V).

A similar result takes place if we use the Yosida approximationΦλinstead ofΦ. Proposition 3.3. Let Φ⊂H×H verifying the hypotheses in Proposition 3.1. Let f ∈ W^1,1(0,T;H) and y0∈D(A)∩D(Φ). Then for allλ >0, there exists a unique strong so- lutiony_N^λ∈W^1,∞(0,T;H)∩L^∞(0,T;D(A))∩C([0,T];V) for problem

d y_N^λ

dt (t) +νAy_N^λ(t) +B_Ny_N^λ(t) +Φλ

y_N^λ(t)=f(t), a.e.t∈(0,T), y_N^λ(0)=y0.

(3.34)

Moreover,y_N^λ is right differentiable, (d⁺/dt)y_N^λ is right continuous, and d⁺

dty_N^λ(t) +νAyN^λ(t) +BNy_N^λ(t) +Φλ

y_N^λ(t)=f(t), ∀t∈[0,T). (3.35) Proof. ΓN=νA+BN+αNI is maximal monotone (forαN large enough),Φλ is demi- continuous monotone, which implies thatνA+BN+Φλ+αNIis maximal monotone in H×H. Then, problem (3.34) has a unique solutiony_N^λ∈W^1,∞(0,T;H) verifying relation (3.35). Moreover, we infer that νAy_N^λ +B_Ny_N^λ +Φλ(y_N^λ)= f −(d y_N^λ/dt)∈L^∞(0,T;H).

Applying (3.3) for y_N^λ(t)∈D(A), we getAy_N^λ ∈L^∞(0,T;H), which implies that y_N^λ ∈ L^∞(0,T;D(A)). Together with (d y_N^λ/dt)∈L^∞(0,T;H), we obtainy_N^λ∈C([0,T];V).

3.2. Estimates for the solution of problem (3.34). ByProposition 3.2, problem (3.2) has a unique strong solution

y_N∈W^1,∞(0,T;H)∩L^∞0,T;D(A)∩C[0,T];V. (3.36)

However, in order to get better estimates, we will further approximate problem (3.2) by problem (3.34), which also has a unique strong solution, byProposition 3.3.

First, we multiply (3.34) byy_N^λ(t) and integrate on (0,t),

t 0

d ds

1

2y_N^λ(s)²

ds+ν ^t

0

Ay_N^λ(s),y_N^λ(s)ds+

t 0

BNy_N^λ(s),y_N^λ(s)ds

+

t 0

Φλ

y_N^λ(s),y_N^λ(s)ds= ^t

0

f(s),y_N^λ(s)ds.

(3.37)

But (BNy_N^λ(s),y_N^λ(s))=0 and (Φλ(y_N^λ(s)),y_N^λ(s))≥(Φλ(0),y_N^λ(s)) becauseΦλis mono- tone,

1

2y_N^λ(t)²+ν ^t

0

y_N^λ(s)²ds

≤ − ^t

0

Φλ(0),y_N^λ(s)ds+1 2y0²+

t 0

f(s),y_N^λ(s)ds≤1 2y0² +1

2

t 0

y_N^λ(s)²ds+1 2

t

02Φ(0)²+f(s)²ds.

(3.38)

In particular, it follows that y_N^λ(t)²≤y0²+ 2

T 0

Φ(0)²+f(s)²ds+

t 0

y_N^λ(s)²ds, (3.39) and by Gronwall’s inequality,

y_N^λ(t)²≤

y0²+ 2

T 0

Φ(0)²+f(s)²ds

e^t. (3.40)

Finally, we infer that 1

2y_N^λ(t)²+ν ^t

0

y_N^λ(s)²ds≤ 1

2y0²+

T 0

Φ(0)²+f(s)²ds

e^t, (3.41) and thus

1

2y_N^λ(t)²+ν ^t

0

y_N^λ(s)²ds≤C1

fL²(0,T;H),y0²

, (3.42)

whereC1is a positive bounded function depending onfL²(0,T;H),|y0|², but indepen- dent ofN,λ.

Next we multiply (3.34) withAy_N^λ(t) and integrate on (0,t):

t 0

d ds

1

2y_N^λ(s)²

ds+ν ^t

0

Ay_N^λ(s)²ds+

t 0

B_Ny_N^λ(s),Ay_N^λ(s)ds

+

t 0

Φλ

y_N^λ(s),Ay_N^λ(s)ds= ^t

0

f(s),Ay_N^λ(s)ds.

(3.43)