REGULARITY CRITERIA FOR WEAK SOLUTIONS OF THE NAVIER-STOKES SYSTEM IN GENERAL UNBOUNDED DOMAINS

. We consider weak solutions of the instationary Navier-Stokes sys-tem in general unbounded smooth domains Ω ⊂ R 3 and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be deﬁned on all types of unbounded domains we have to replace the space L q (Ω), q > 2, by ˜ L q (Ω) = L q (Ω) ∩ L 2 (Ω) and Serrin’s class L r (0 ,T ; L q (Ω)) by L r (0 ,T ; ˜ L q (Ω)) where 2 < r < ∞ , 3 < q < ∞ and 2 r + 3 q = 1.


Dedicated to our colleagues Paolo Secchi and Alberto Valli on the occasion of their 60th birthdays
Abstract. We consider weak solutions of the instationary Navier-Stokes system in general unbounded smooth domains Ω ⊂ R 3 and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be defined on all types of unbounded domains we have to replace the space L q (Ω), q > 2, bỹ L q (Ω) = L q (Ω) ∩ L 2 (Ω) and Serrin's class L r (0, T ; L q (Ω)) by L r (0, T ;L q (Ω)) where 2 < r < ∞, 3 < q < ∞ and 2 r + 3 q = 1.
1. Introduction. We consider the instationary Navier-Stokes system u t − ∆u + div(u ⊗ u) + ∇p = f in (0, T ) × Ω, div u = 0 in (0, T ) × Ω, u = 0 on (0, T ) × ∂Ω, u(0) = u 0 at t = 0, in a general unbounded domain Ω ⊂ R 3 with uniform C 2 -boundary and a finite time interval (0, T ). Here u = (u 1 , u 2 , u 3 ) denotes the unknown velocity field, p an associated pressure, f a given external force of the form f = f 1 + div f 2 , and u 0 the initial value of u at time t = 0. For simplicity, the viscosity is set to ν = 1. A precise definition of domains with uniform C 2 -boundary can be found in Definition 2.1 below. A problem in this setting is the unboundedness of the underlying domain Ω. Due to counter-examples by M.E. Bogovskij and V.N. Maslennikova [2,3] the Helmholtz decomposition of vector fields in L q (Ω), 1 < q < ∞, on an unbounded smooth domain may fail unless q = 2. By analogy, a bounded Helmholtz projection P q with the properties required to define the Stokes operator A q = −P q ∆ when q = 2 may not exist. Therefore, in [5,7,8,9,10] H. Kozono, H. Sohr and the first author of this article introduced the spaces L q (Ω) := L q (Ω) + L 2 (Ω), if 1 ≤ q < 2, L q (Ω) ∩ L 2 (Ω), if 2 ≤ q ≤ ∞.
Our first result shows that a weak solution in the sense of Leray and Hopf must coincide with a very weak solution; for the definition and existence of very weak solutions see Definition 2.2 and Theorem 2.3 in Sect. 2 below. The negative Sobolev space T −1,r,q (T, Ω) will also be explained in Sect. 2, cf. (18). This result will be important for showing regularity of weak solutions.
Furthermore, assume that the data functional F defined by lies in T −1,r,q (T, Ω) and that there exists a very weak solution u ∈ L r (0, T ;L q σ (Ω)) of the Navier-Stokes equations to this data sharing the additional important property u ∈ L 4 (0, T ; L 4 (Ω)).
Assume data u 0 ∈ L 2 σ (Ω) and and that u is a weak solution in the sense of Leray and Hopf satisfying the strong energy inequality for a.a. s ∈ (0, T ) (including s = 0) and all s ≤ t < T . Moreover, choose exponents 1 ≤ r 0 ≤ r and 3 < q 0 ≤ q. Then there exists a constant η = η(Ω, q, r, q 0 , r 0 , T ) > 0 with the following property: If for a point t ∈ (0, T ) lim inf where α = r0 2 2 r0 + 3 q0 − 1 , then t is a regular point. In the case where q 0 = q this condition reads lim inf and is even a necessary condition.
In particular, the left-side Serrin-condition u ∈ L r (t − δ, t;L q (Ω)) for some δ > 0 is sufficient for t to be a regular point.
Actually, the constant η depending on the domain Ω only depends on the socalled type of the domain (type(Ω)); this notion will be explained in Sect. 2.
A consequence of Theorem 1.2 is a criterion based on the kinetic energy E kin (t) = , and let u be a weak solution satisfying the strong energy inequality (for simplicity we assume There is a constant η = η(type(Ω), T ) with the following property: If for a point t ∈ (0, T ) and some µ > 0 the left-sided 1 2 -Hölder continuity holds, then t is a regular point of u in the sense u ∈ L 4 (t − δ, t + δ;L 6 (Ω)).
there are constants α, β, K > 0 such that for all x 0 ∈ ∂Ω there exist -after an orthogonal and an affine coordinate transform -a function h on the closed ball B α (0) ⊆ R n−1 of class C 2 and a neighborhood U α,β,h (x 0 ) of x 0 with the following properties: h C 2 ≤ K and h(0) = 0, h (0) = 0; moreover, The triple (α, β, K) is called the type of Ω and will be denoted by type(Ω) = (α, β, K). For a constant C in some estimate we will write C = C(type(Ω)) if it does depend only on α, β and K, but in no other way on Ω.
The spaceL q (Ω) admits the direct algebraic and topological decompositioñ L q (Ω) =L q σ (Ω) ⊕G q (Ω) yielding a projectionP q fromL q (Ω) ontoL q σ (Ω) with operator norm bounded by a constant c = c(q, type(Ω)), see [7]. We have the duality relations P q * =P q andL q σ (Ω) * =L q σ (Ω). Using the Helmholtz projectionP q we can define the Stokes operatorÃ q , 1 < q < ∞, for a uniform C 2 -domain Ω ⊆ R n with domain where D q := L q σ (Ω) ∩ W 1,q 0 (Ω) ∩ W 2,q (Ω). ThenÃ q := −P q ∆ : D(Ã q ) ⊆L q σ (Ω) → L q σ (Ω) is a densely defined closed operator inL q σ (Ω) satisfying Ã q * =Ã q and generating an analytic semigroup e −tÃq , t ≥ 0, with bound for all f ∈L q σ (Ω) and t ≥ 0 with a constant C = C(q, δ, type(Ω)), δ > 0, see [10]. It is unknown whether the usual resolvent estimate for the infinitesimal generatorÃ q of the semigroup e −tÃq holds uniformly in the resolvent parameter λ in a sector of C as |λ| → 0. Therefore, the semigroup may increase exponentially fast. Note that from time to time we will omit the symbols Ω and T for domain and length of the time interval, respectively, when this data is known from the context.
For an external force f ∈ L r (0, T ;L q σ (Ω)) and an initial value u 0 ∈ D(Ã q ) (for simplicity) consider the abstract Cauchy problem It is known that there exists a unique solution u ∈ L r (0, T ; D(Ã q )) ∩ W 1,r (0, T ; L q σ (Ω)) which can be represented by the variation of constants formula Moreover, it satisfies the maximal regularity estimate A further crucial property of the Stokes operator is the fact that 1 +Ã q admits bounded imaginary powers, see [14]. Hence complex interpolation methods can be used to describe domains of fractional powers (1 +Ã q ) α . For 0 ≤ α ≤ 1 let the domain of (1 +Ã q ) α be denoted bỹ equipped with the norm (1+Ã q ) α · Lq . If −1 ≤ α < 0 defineD α q as the completion ofL q σ (Ω) in the norm (1 +Ã q ) α · Lq . These spaces are reflexive and satisfy the duality relation (D α q ) * =D −α q . As special cases we get that . This result implies the following embedding and decay estimate ([15, Proposition 3, Theorem 1]): Let 1 < q ≤ r < ∞, and α := 3 for every u ∈D α q and f ∈L q σ (Ω), respectively, and for any t > 0 and δ > 0; here For a discussion of spaces of initial values in Proposition 2 below it is reasonable to consider also Lorentz spaces overL q (Ω) and their solenoidal subspaces. First we define for 1 < q < ∞, 1 ≤ ρ ≤ ∞ the Lorentz spaces letting the case q = 2 undefined; here L q,ρ (Ω) denotes a usual Lorentz space, cf.
(ii) For more concrete conditions on the data functional F, including the higher dimensional case n ≥ 3 and the case of nonhomogeneous boundary data as well as nonsolenoidal velocity fields, we refer to Propositions 2.4, 3.4, 4.5, and Corollary 4.6 in [11].
then there exists a unique very weak solution u ∈ L r (0, T ;L q (Ω)) to the Navier-Stokes system with data F in the sense of Definition 2.2. The a priori estimate holds with a constant C = C(type(Ω), q, T ).
In the case of more regular data u 0 , f 1 , f 2 for F as in (17) the very weak solution u has the integral representation (variation of constants formula) (22) is defined in the sense of distributions (with solenoidal vector fields as test functions) For the application to questions of regularity we need that the very weak solution in Theorem 2.3 is contained in L 4 (0, T ; L 4 (Ω)) as well. The following Proposition describes conditions on the data under which this property will hold.
Proof. The result is a special case of [11,Corollary 4.6]. The main idea of the proof is to show that not only F ∈ T −1,r,q (T, Ω), but also F ∈ T −1,4,4 (T, Ω), and to apply [11,Theorem 4.4].
The "optimal" condition in terms of real interpolation theory is In particular, the conditions u 0 ∈L ρ σ (Ω) and 3. Proofs and further results.

Identification of weak and very weak solutions.
It is well-known that Serrin's condition u ∈ L r loc (0, T ; L q (Ω)), 2 < r < ∞, 3 < q < ∞, 2 r + 3 q = 1, is a sufficient condition for a weak solution u in the sense of Leray and Hopf to be unique and regular, see e.g. [16, Ch. V, Theorems 1.5.1, 1.8.1, 1.8.2]. Therefore, the following definition is useful: Definition 3.1. Let u be a weak solution to the Navier-Stokes system on Ω×(0, T ) in the sense of Leray and Hopf. Then a point t ∈ (0, T ) is called regular point of u if there exist δ > 0 (with δ ≤ min(t, T − t)) and exponents 2 < r < ∞ and 3 < q < ∞ with 2 r + 3 q = 1 such that u ∈ L r (t − δ, t + δ;L q (Ω)). Proof of Theorem 1.1. We will show that the very weak solution u is also a weak solution in the sense of Leray and Hopf. Using Serrin's uniqueness criterion we conclude that u andũ coincide almost everywhere.
To finish the proof we remark that u as very weak solution is contained in Serrin's uniqueness class L r (0, T ; L q ). Since the weak solutionũ satisfies the energy inequality, due to [16, Ch. V, Theorem 1.5.1], u coincides withũ.
This theorem can be used to prove the short-time existence and uniqueness of regular (strong) solutions.
It is clear that u ∈ L r (0, T ;L q σ (Ω)) for all exponents (r, q) as in the formulation of the corollary.

3.2.
Local and global regularity criteria. First we present an abstract result which will be the key to all regularity criteria in the sequel.
As an application we get a result which can be considered as extension of Serrin's uniqueness theorem. It generalizes the uniqueness result for weak solutions in the class L ∞ (0, T ; L 3 (Ω)) of [12,13] from bounded domains to general unbounded domains.
By the weak continuity of both u and v with values in L 2 σ (Ω), it follows that u(T * ) = v(T * ) and -by weak continuity of v with values inL 3 σ (Ω) -that even u(T * ) = v(T * ) ∈L 3 σ (Ω). Since u satisfies the strong energy inequality we find (t j ) j ⊂ (0, T * ) with t j T * such that for all t ≥ T * . For the term 1 2 u(t j ) 2 2 we get, using the energy equality for v, that 1 2 Hence, taking the limit j → ∞, we find the energy inequality for all t > T * . Now we again use Lemma 3.3 to find some δ 1 > 0 such that u = v ∈ L r (T * , T * + δ 1 ;L q σ (Ω)) contradicting the definition of T * . Hence T * = T . Finally, since v satisfies the energy equality and v(t) ∈L 3 σ (Ω) for all t, Lemma 3.3 proves that u = v ∈ L r (t, t+δ(t);L q σ (Ω)) for all t ∈ [0, T ) and some δ(t) > 0.
A slight modification of the above proof shows the following result: Under the basic assumptions of Theorem 3.4 let u and v be weak solutions to the Navier-Stokes equations with initial data u 0 ∈ L 2 σ (Ω). Assume that u satisfies the strong energy inequality and that v(t) ∈L 3 σ (Ω) or v(t) ∈D
Then u = v a.e. and u = v ∈ L r (0, T ;L q σ (Ω)). Now we derive more regularity criteria. The next theorem shows that regularity from the left already implies regularity.
Proof of Theorem 1.2. First we prove sufficiency. Let η > 0 be the constant from Lemma 3.2 and choose δ 0 > 0 small enough to fulfill the last two conditions in Lemma 3.2, i.e., Next, for every δ > 0 we find t 0 ∈ (t − δ, t) such that u satisfies the energy inequality starting at t 0 , u(t 0 ) ∈L q0 σ (Ω) and Moreover, let t 1 = t + δ. Now we use theL r -L q -estimate (15) to find that , since t + δ − τ ≤ 2δ. By assumption (6) we find 0 < δ ≤ δ 0 such that the right-hand term in the last inequality above is smaller than η , if only the new constant η > 0 is chosen small enough (set η := η r0/r /C). This proves that for this choice of t 0 and t 1 the first condition in Lemma 3.2 is satisfied. Since δ ≤ δ 0 also the second and third condition is fulfilled. Thus Lemma 3.2 proves regularity of the point t.
Next we prove that the condition (6) is necessary in case q 0 = q. Let t be a regular point of u. Then, for any 1 ≤ r 0 ≤ r we get by Hölder's inequality for δ → 0+. This proves that the condition (6) is necessary in case q 0 = q. It is only left to show that u ∈ L r (t − ε , t;L q ) is sufficient for t to be regular. This is easily seen by choosing q 0 = q and r 0 = r. Now α = 0, and (6) reads lim inf which is obviously satisfied by the assumption in view of Lebesgue's theorem on dominated convergence. This finishes the proof of Theorem 1.2.
The next global regularity result is a consequence of Theorem 1.2.
Hence we proceed to get that and u r0 L r 0 (t−δ1/2,t;L q 0 ) ≤ η r0 u 0 αr0 L q 1 by assumption, we summarize the above estimates and get t1−t0 0 e −τÃq u(t 0 ) r L q dτ ≤ C 2 η r , with a fixed constant C 2 = C 2 (type(Ω), q, r, q 0 , q 1 , r 0 , T ). Hence, if the constant η > 0 is chosen small enough we can apply Lemma 3.2 to find that t is a regular point. Now a compactness argument finishes the proof.
3.3. Energy criteria. From Theorem 1.2 we can also derive regularity conditions in terms of u 2 or ∇u 2 , i.e., in terms describing physical energies. For simplicity we assume f 1 = 0 and f 2 = 0.
Theorem 3.5. Let Ω ⊆ R 3 be a uniform C 2 -domain, 0 < T < ∞. Assume that u with initial data u 0 ∈ L 2 σ (Ω) is a weak solution satisfying the strong energy inequality.
There is a constant η = η(type(Ω), T ) such that the following holds: If for a point t ∈ (0, T ) it holds that lim inf then t is a regular point of u in the sense that u ∈ L 4 (t − ε, t + ε;L 6 (Ω)).
Proof. In Theorem 1.2 we choose q = q 0 = 6, r = 4, r 0 = 2. The constant for smallness in (6) will here be denoted by η so that we have to show that lim inf We estimate by Sobolev's embedding theorem u(τ ) L6 ≤ u(τ ) L 2 + C ∇u(τ ) L 2 and find 1 Here the first term can be estimated using the energy equality by δ 1/2 u 0 2 L 2 which tends to zero for δ → 0. The second term is smaller than η /2 for small δ > 0 if only the new constant η > 0 is chosen small enough. So Theorem 1.2 implies the result.
Proof of Theorem 1.3. To use Theorem 3.5 let η > 0 be the constant from that theorem and η := η . Choose a sequence δ k 0 as k → ∞ with the property that u satisfies the energy estimate with starting point t − δ k for all k ∈ N. For this sequence we can estimate As k → ∞ the right-hand side is bounded by η = η so that the preceding theorem shows that t is a regular point. This finishes the proof.