Navier-Stokes Equations with Potentials

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.

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 solutions are stated in Section 2. The first of these theorems is proved in Section 3 and the others in Section 4. Section 5 is concerned with weak solutions.The last section is devoted to examples.

Abstract and Applied Analysis
We will use the standard spaces (see, e.g., [1][2][3]) n ; div y = 0 in Ω, y • n ∂Ω = 0 on ∂Ω , H is a real Hilbert space endowed with L 2 -norm | • | and V is a real Hilbert space endowed with H 1 0 -norm • = |∇ • |.Moreover, denoting by V the dual space of V and considering H identified with its own dual, we have V ⊂ H ⊂ V algebraically and topologically with compact injections.
Here, (•,•) denotes the scalar product of H and the pairing between V and its dual V .The norm of V is denoted by • V .
Let A ∈ L(V ,V ) (the space of linear continuous operators from V in V ), (Ay,z) = n i=1 Ω ∇y i • ∇z i dx, for all y,z ∈ V .We have (Ay, y) = y 2 , for all y ∈ V .We set D(A) = {y ∈ V ; Ay ∈ H} and denote again by A the restriction of A to H.
Adriana-Ioana Lefter 3 Suppose Φ satisfies the following hypotheses: (h 1 ) Φ = ∂ϕ, where ϕ : where We consider the classical definition of the maximal monotone operator.We will denote In the sequel, the symbol will be used to denote convergence in the weak topology, while the strong convergence will be denoted by →.

Main results for strong solutions
Theorem 2.1.Let T > 0 and let Ω ⊂ R n , n = 2,3 be an open and bounded domain, with a smooth boundary.Assume that Moreover, y is right differentiable, (d + /dt)y is right continuous, and If n = 3, the solution y exists on some interval [0,T 0 ), where We have denoted by y → (νAy + By + Φ(y) − f (t)) 0 the minimal section of the multivalued mapping y → (νAy + By + Φ(y) − f (t)).
If we ask for lower regularity of the initial data, we obtain the following results.
has a unique solution Remark 2.4.We obtain the same results if Φ satisfies the following hypotheses: (H 1 ) Φ is a single-valued maximal monotone operator in H × H; (H 2 ) there exist three constants γ 1 ,γ 2 ≥ 0, and α ∈ (0,ν) such that In the sequel, we use the same symbol C for various positive constants.

Proof of Theorem 2.1
The proof uses the theory of nonlinear differential equations of accretive type in Banach 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 nonlinearity B with a truncation B N , N ∈ N * .We may then state existence and uniqueness results for the approximate equations (3.2), (3.34) involving B N , Φ, and B N , Φ λ , λ > 0 instead of B, Φ (Propositions 3.2, 3.3).We intend to prove that for N large enough, the solution of the truncated problem involving B N , Φ coincides with the solution of the initial problem.To this aim, we need to obtain estimates on the solution y N of problem (3.2).In order to do this, we are obliged to deduce the convenient estimates first on problem (3.34) (the one involving Φ λ ) because relation (1.12) does not extend in a suitable way to arbitrary elements of Φ(y N (t)).Passing to the limit with λ → 0 in (3.34), we return to the problem in B N , Φ and conclude the proof.

Approximate problems: existence and uniqueness.
For N ∈ N * , define the modified nonlinearity B N : Moreover, there exists a constant C N > 0 such that ) Proof.It has been proved in [4] (see Lemma 5.1, page 292) that νA + B N applies D(A) into H and that for α N large enough, the operator ) and Λ N = Γ N + Φ is the sum of two monotone operators, and by consequence it is a monotone.In order to obtain the maximal monotony of Λ N , it is sufficient to prove that R(I + Λ N ) = H.Let f ∈ H and λ > 0 a fixed.We approximate the equation by the equation that is where Φ λ is the Yosida approximation of Φ.By the properties of the Yosida approximation, Φ λ is demicontinuous monotone and its sum with the maximal monotone operator Γ N is maximal monotone, which implies the existence of a solution u λ ∈ D(A) for (3.6).
The uniqueness follows by monotony arguments.Let μ N = α N + 1; then (3.6) reads We first multiply (3.8) by u λ and infer that ), and we have Consequently, where the constant C > 0 does not depend on λ.
For the first one, we consider λ > 0 fixed, w ∈ D(A), and let g λ = νAw+B N w + Φ λ (w) + μ N w.In the same way as we deduced (3.23), we may obtain |Aw| ≤ C(1 where the constant C > 0 does not depend on λ.Thus (3.3) is proved.
In order to prove the second relation, we take w ∈ D(A) ∩ D(Φ) and η ∈ Φ(w).Let g = νAw + B N w + η + μ N w.For this g, we may construct as in the first part of the proof a sequence (w λ ) λ>0 ⊂ H such that Moreover, w λ → w, Aw λ Aw because Λ N is maximal monotone.Passing to the limit with λ → 0 in (3.23) written for (w λ ), we obtain |Aw| ≤ C(1 which proves relation (3.4).This concludes the proof of Proposition 3.1.

Estimates for the solution of problem (3.34)
. By Proposition 3.2, problem (3.2) has a unique strong solution (3.37) In particular, it follows that and by Gronwall's inequality, 2 ds e t . (3.40) Finally, we infer that and thus where C 2 , C 2 are positive bounded functions of f L 2 (0,T;H) , |y 0 | 2 , y 0 2 , but do not depend on N,λ.
We get The idea of the proof is to approximate the initial data with sequences of functions satisfying the hypotheses of Theorem 2.1 and then to pass to the limit.Let (y According to Theorem 2.1, problem dy j (t) dt + νAy j (t) + By j (t) + Φ y j (t) f j (t), a.e.t ∈ (0,T), has a unique solution y j ∈ W 1,∞ (0,T 0 ;H) ∩ L ∞ (0,T 0 ;D(A)) ∩ C([0,T 0 ];V ), where T 0 = T if n = 2 and T 0 ≤ T in n = 3.Moreover, y j satisfy the estimates The constants are independent of j.

Adriana-Ioana Lefter 21
Consequently, y j ) j is bounded in C 0,T 0 ;V ∩ L 2 0,T 0 ;D(A) , Ay j j , By j j , η j j are bounded in L 2 0,T 0 ;H , dy j dt j is bounded in L 2 0,T 0 ;H .Then, on a subsequence again denoted by (y j ) j , we have for j → ∞, Passing to the limit with j → ∞, we prove the existence of the strong solution.
In order to prove the uniqueness of the solution, we proceed as in the proof of Theorem 2.1 in Section 3.4.

Weak solutions
Consider the operator Φ : V → V monotone and demicontinuous.From the definition of demicontinuity, we infer that Φ is also single valued and its domain is We will denote by the same symbol Φ the operator Φ : V → V and its restriction from Assume in addition that (h 1 ) 0 ∈ D(Φ); (h 2 ) there exist two constants γ ≥ 0, α ∈ (0,1/ν) such that where Φ λ is the Yosida approximation of Φ ⊂ H × H; (h 3 ) there exist p ≥ 2, ω 1 ,ω 2 > 0, μ ≥ 0 constants such that 3) The following result on weak solutions takes place.(5.5) The weak solution is unique if n = 2.
Proof.First we will fix N ∈ N * and we will prove that problem has a unique solution L r (0,T;V ) ifn = 3. (5.7) Then we will pass to the limit with N → ∞.Let (y j 0 ) j∈N ⊂ D(A) ∩ D(Φ) and ( f j ) j∈N ⊂ W 1,1 (0,T;H) such that Thus we may apply the existence and uniqueness theorems for strong solutions in Section 2. Then (see [7]), problem where z(t) = y(t) − y 1 , has finite time extinction property, that is, z(T) = 0 for ρ large enough.
Example 6.2 (invariance preserving (see [4]) where N K (y) = {z ∈ H; (z, y − x) ≥ 0, for all x ∈ K} is the Clarke normal cone to K at y.The operator Φ(y) = N K (y) coincides with the subdifferential (∂I K )(y) of the indicator function (6.12) Then hypotheses (h 1 )-(h 3 ), Section 1 on Φ, are verified by ∂I K = N K and we may apply the results in Section 2.
Example 6.3 (stabilizing feedback controllers).Consider the controlled system dy dt (t) + νAy(t) + By(t) = f e + u(t), t > 0, y(0) = y 0 . (6.13) Let y e ∈ D(A) be a steady-state solution for (6.13), that is, y e verifies νAy e + By e = f e .(6.14) Given K ⊂ H a closed and convex set, with 0 ∈ K, we look for a feedback controller u such that y(t) − y e ∈ K, for all t ≥ 0 and lim t→∞ |y(t) − y e | = 0.
We set z = y − y e and take u(t) ∈ −λz(t) − (∂I K )(z(t)), with λ > 0 large enough.Then (6.In the proofs of Theorems 2.1, 2.2, 2.3, the operator B will be replaced by B + A 0 .This fact does not change the estimates in Section 3. Of course, the positive constants in the right-hand side of the estimates will depend on y e , |Ay e |.Although, unlike B, (A 0 w,w) = b(w, y e ,w) = 0 in V , the resulting terms are absorbed by other terms.