On the uniqueness of a solution to a stationary convection-diffusion equation with a generalized divergence-free drift
aa r X i v : . [ m a t h . A P ] J un On the uniqueness of a solution to a stationaryconvection-diffusion equation with a generalizeddivergence-free drift
M.D. Surnachev ∗ Computational Aeroacoustcs Laboratory, Keldysh Institute of Applied Mathematics RASMiusskaya Sq. 4, Moscow 125047, Russia
Abstract
Let A be a skew-symmetric matrix in L (Ω), Ω — a bounded Lipschitz do-main in R n , n ≥
2. The Dirichlet problem − div ( ∇ u + A ∇ u ) = f , u ∈ H (Ω), f ∈ W − , (Ω) has at least one solution obtained by approximating A and passingto the limit. In 2004 V.V. Zhikov constructed an example of nonuniqueness. In thesame paper he proved the uniqueness of solutions if the L p (Ω) norms of A are o ( p )as p goes to infinity. We prove the uniqueness of solutions if exp( γ | A | ) ∈ L (Ω) forsome γ >
0, which generalizes Zhikov’s theorem.
Keywords : uniqueness; generalized drift; BMO; Morrey space.
MSC2010 : 35J15.
Dedicated to the memory of Academician V.I. Smirnov,One of the Founding Fathers of MathPhys in Russia
Let Ω be a bounded Lipschitz domain in R n , n ≥ f an element of W − , (Ω) and A askew-symmetric matrix from L (Ω). In this paper we are concerned with the question ofuniqueness of solutions to the Dirichlet problem Lu = − div ( ∇ u + A ∇ u ) = f, u ∈ W , (Ω) . (1)By a solution we mean a function u ∈ W , (Ω) such that the integral identity Z Ω ( ∇ u + A ∇ u ) ∇ ϕ dx = ( f, ϕ ) (2)holds for any ϕ ∈ C ∞ (Ω).Let us elucidate the term “generalized drift” in the paper title. Formally,( A ij u x j ) x i = A ij u x i x j + u x j A ij,x i = ( uA ij,x i ) x j − uA ij,x i x j = ( uA ij,x i ) x j . Here and below we use the Einstein convention of summation over repeated indices. Forscalar and vector functional spaces we use the same notation, i.e. for A : Ω → R n × n ∗ Email: [email protected]
1e write A ∈ L (Ω) instead of A ∈ L (Ω) n × n , for a vector field a : Ω → R n we write a ∈ L (Ω) instead of a ∈ L (Ω) n etc.More rigorously, if A ∈ W , (Ω) and ϕ ∈ C ∞ (Ω), h− div ( A ∇ u ) , ϕ i = Z Ω A ij u x j ϕ x i dx = − Z Ω A ij,x j uϕ x i dx − Z Ω uA ij ϕ x i x j dx = Z Ω A ji,x j uϕ x i dx = Z Ω ( u div A ) ∇ ϕ dx = h− div ( u div A ) , ϕ i , where (div A ) i = A ji,x j . Since A is skew-symmetric, div div A = A ij,x i x j = 0. Thus, for askew-symmetric A ∈ W , (Ω) the Dirichlet problem (1) can be written in the form − div ( ∇ u + a ∇ u ) = f, u ∈ W , (Ω) , f ∈ W − , (Ω) , (3)with the solenoidal vector field a = div A ( a i = A ji,x j ). For nonsmooth A one can say[1], [2] that (1) describes “diffusion in a turbulent flow” (in our case, stationary) sincethe flow velocity a = div A exists only in the sense of distributions. A similar class ofequations in “generalized divergence form” was studied in [3].On the other hand, given a (smooth) solenoidal vector field a we can construct (atleast, locally) a skew-symmetric matrix A such that a = div A . Indeed, solenoidal a corresponds to the closed ( n −
1) form ω = ∗ ( a i dx i ) ( ∗ — the Hodge star operator).By the Poincar´e lemma (for instance, [4]) it is also exact, ω = dα for ( n −
2) form α ,provided that Ω is star-shaped (or contractible to a point, or diffeomorphic to a ball).The coefficients of α give the coefficients of A . In the language of differential forms, thepassage between (1) and (3) is equivalent to the relation R ∂D udα = ( − n − R ∂D α ∧ du , D a subdomain of Ω, which follows from d ( udα + ( − n α ∧ du ) = 0.The form α can be additionally normed by δα = 0 ( δ — codifferential), and soughtin the form α = δβ , which eventually leads to the problem ( dδ + δd ) β = ω with suitableboundary conditions. For the problem a = div A the condition δα = 0 is equivalent to A ij,x k + A jk,x i + A ki,x j = 0, which is neccesary for the representation of A in the formof the rotor of a vector field V , i.e. A ij = curl ij V = V i,x j − V j,x i . More on the Hodgedecomposition for differential forms can be found in the famous Morrey’s monography[5, Chapter 7]. A rather complete theory of differential forms on Lipschitz domain wasconstructed in [6] in the framework of Besov spaces.In dimension 2 this reduces to A = (cid:18) α − α (cid:19) , div A = ( − α y , α x ) = ( a , a ) . Since a is solenoidal the vector field V = ( a , − a ) is potential. So one needs to find afunction α with the given gradient V . In other words a = ∇ ⊥ α .In dimension 3, any skew-symmetric matrix can be represented as Ax = w × x , andthe problem of finding A such that a = div A reads as rot w = a , which is also easy tosee from div ( w × ∇ u ) = ∇ u · rot w − w · rot ∇ u = div ( u rot w ) . (4)The problem of finding a vector field with prescribed rotor (and divergence) is a classicalproblem of vector calculus. For Ω = {| x | < R } one of solutions obtained by the Poincar´elemma is w ( x ) = R a ( tx ) × tx dt . 2f Ω = R n and the solenoidal vector field a is vanishing at infinity, a solution todiv A = a can be obtained as the curl of the newtonian potential of a : A ij = V i,x j − V j,x i , V i ( x ) = ( n ( n − ω n ) − Z R n a i ( y ) | x − y | − n dy, n ≥ , (5)where ω n is the volume of the unit ball in R n , with the obvious modification for n = 2. IfΩ is a bounded domain and the normal component of a on the boundary of Ω is equal tozero, then a solution to div A = a is given by the same formula (5), where a is extendedby zero outside Ω (this extension is also solenoidal).In dimension 3 formula (5) represents the standard vector calculus solution to rot w = a defined as w = rot (4 π ) − R R n a ( y ) | x − y | − dy , which follows from representing w = rot v and using the vector calculus identity rot rot v = ∇ div v − △ v . Such representation isof course only possible under the condition div w = 0, which is equivalent to requiring δα = 0 above.If the normal component of a on the boundary is not equal to zero, one can continue a to a sufficiently large ball B which contains Ω by solving the auxilliary Neumann problem −△ u = 0 in B \ Ω, ∂u/∂n = a · n on ∂ Ω, ∂u/∂n = 0 on ∂B ( n — the exterior unitnormal to B \ Ω). Then one sets a = ∇ u in B \ Ω, a = 0 in R n \ B , and a solution todiv A = a is given by (5). This construction assumes that either Ω does not have holes,or the flow of a across the boundary of each hole is zero.If Ω has holes, the representation a = div A is obviously not always possible, but bythe Hodge (Weyl in 3D) theorem there exists a harmonic (irrotational solenoidal) vectorfield b such that a = b + div A . For instance, one can take b = P j c j ∇ Γ( x − x j ) where x j is a point inside the j -th hole, Γ is the fundamental solution of the Laplace, and theconstants c j are chosen to balance the flux of a across the boundary of the correspondinghole. In detail this construction is discussed in [15].Another way is to directly solve the problem −△ V = a in Ω, V × n = 0 and div V = 0on ∂ Ω, and find A = curl V . Regarding the equation rot u = f and correspondingboundary value problems see [7] (classical potential theory), [8] (modern potential theory)and recent papers [9, 10] (Galerkin’s method). For the closely related problem of findinga solenoidal vector field with prescribed boundary value (or a vector field with givendivergence) we refer the reader to [11, 12, 13, 14]. It is easy to prove that (1) has at least one solution. Indeed, take a sequence A N ofbounded skew-symmeric matrices converging to A in L (Ω). Let u N be solutions to thecorresponding problems − div ( ∇ u n + A N ∇ u ) = f, u N ∈ W , (Ω) , i.e. Z Ω ( ∇ u N + A N ∇ u N ) ∇ ϕ dx = ( f, ϕ ) for all ϕ ∈ C ∞ (Ω) .
3y the Lax-Milgram lemma such solutions exist and are uniqely defined. Using thetest-function u N in the corresponding integral identity, we have Z Ω |∇ u N | dx = ( f, u N ) , (6)wherefrom Z Ω |∇ u N | dx ≤ Z Ω f dx. Extracting from u N a weakly convergent in W , (Ω) subsequence and passing to the limitin the integral identity Z Ω ( ∇ u N + A N ∇ u N ) ∇ ϕ dx = ( f, u N ) , we obtain a solution u to (1). Passing to the limit in (6) we see that this solution satisfiesthe energy inequality Z Ω |∇ u | dx ≤ ( f, u ) . (7)Following Zhikov [16] we call a solution constructed by this procedure an appoximationsolution. In the same paper V.V. Zhikov constructed an example of nonapproximationsolutions, which satisfy the “unnatural” energy inequality Z Ω |∇ u | dx > ( f, u ) . (8)Denote [ u, ϕ ] = Z Ω A ∇ u ∇ ϕ, so that (2) can be rewritten as Z Ω ∇ u ∇ ϕ dx + [ u, ϕ ] = ( f, ϕ ) . (9)It is clear that | [ u, ϕ ] | ≤ C k∇ ϕ k L (Ω) for all ϕ ∈ C ∞ (Ω) , (10)and [ u, ϕ ], initially defined for ϕ ∈ C ∞ (Ω), can be extended to a linear bounded functionalon W , (Ω). Accordingly, in (9) the set of admissible test functions can be extended to W , (Ω). Substituting u as a test function in (9) we obtain Z Ω |∇ u | dx + [ u, u ] = ( f, u ) . (11)On the other hand, any u ∈ W , (Ω) satisfying (10), is a solution to (1) with theright-hand side f defined by (9). So, the set of functions u ∈ W , (Ω) satisfying (10) isthe set of all solutions to (1) when f ranges over W − , (Ω). For a given skew-symmetricmatrix A we denote this set by D ( A ). When necessary to distinguish between differentmatrices, we add a subscript to [ · , · ]: for instance, [ u, v ] A .4he rest of this section is devoted to certain elementary observations. Inequality (7)translates into [ u, u ] ≥ u, u ] <
0. Since an approximation solution always exists this immediatelyimplies nonuniqueness. On the other hand, if [ u, u ] ≥ u ∈ W , (Ω) then for anyright-hand side f a solution is unique, and (7) holds. Another easy observation is that[ u, u ] = 0 for all u ∈ W , (Ω) is equivalent to the uniqueness of solutions together withthe energy identity Z Ω |∇ u | dx = ( f, u ) (12)for all f .Also note that if there exists u with [ u, u ] >
0, then for problem (1) with A replacedby − A there exists a nonapproximation solution. Analogously, if for a given matrix A there exists a solution with [ u, u ] < A replaced by − A thereexists a solution which satisfies the strict energy inequality Z Ω |∇ u | dx < ( f, u ) . If for some right-hand side f there exist multiple solutions, then there exists a nontrivialsolution u corresponding to f = 0. For u identity (11) gives[ u , u ] = − Z Ω |∇ u | dx < . Since for any solution u there holds[ u + tu , u + tu ] = [ u, u ] + t [ u , u ] + t [ u, u ] + t [ u , u ] < t, (13)then nonuniqueness for some f implies nonuniqueness for all right-hand sides f .The same observation also allows us to single out an “extremal” solution from L − f .Indeed, consider I [ f ] = sup { [ u, u ] , u ∈ L − f } . Since an approximation solution alwaysexists, 0 ≤ I [ f ]. For solutions, satisying [ u, u ] ≥ k u k ≤ k f k and [ u, u ] ≤ ( f, u ) ≤ k f k .It follows that I [ f ] ≤ k f k . Take a sequence u k ∈ L − f such that [ u k , u k ] monotonicallyincreases and converges to I [ f ]. Then one can easily verify that12 Z Ω |∇ ( u k − u m ) | dx = 2 (cid:20) u k + u m , u k + u m (cid:21) − [ u k , u k ] − [ u m , u m ] → k, m → ∞ . Thus, u k → u strongly in W , (Ω), Lu = f and[ u, u ] = ( f, u ) − Z Ω |∇ u | dx = lim k →∞ ( f, u k ) − Z Ω |∇ u k | dx = lim k →∞ [ u k , u k ] = I [ f ] . For any z ∈ L − t ∈ R we have u + tz ∈ L − f , so [ u + tz, u + tz ] ≤ [ u, u ]. Hence u satisfies [ u, z ] + [ z, u ] = 0 for any z ∈ L − . (14)From (13), any function from L − f satisfying the latter property maximizes I [ f ] and isuniquely defined. Denote the special solution of Lu = f which maximizes I [ f ] by ˜ L − f .5t is obvious that ˜ L − f + ˜ L − g ∈ L − ( f + g ) and satisfies (14). Therefore ˜ L − f + ˜ L − g =˜ L − ( f + g ). So, ˜ L − f : W − , (Ω) → W , (Ω) is a linear bounded operator, which is theright inverse for L .For any skew-symmetric matrix B ∈ L ∞ (Ω) there holds | [ u, ϕ ] | ≤ C k B k ∞ k∇ u k L (Ω) k∇ ϕ k L (Ω) , which implies D ( B ) = W , (Ω) , [ u, u ] B = Z Ω B ∇ u ∇ u dx = 0 . Thus, addition of any skew-symmetric matrix B ∈ L ∞ (Ω) to A does not change D ( A )and [ u, u ]: D ( A + B ) = D ( A ) , [ u, u ] A + B = [ u, u ] A + [ u, u ] B = [ u, u ] A . In certain sense, the information on uniqueness/nonuniqueness is contained in the set oflarge values of A . In [16] Zhikov proved the following Theorem (Zhikov) . Let lim p →∞ p − k A k L p (Ω) = 0 . (15) Then (1) has a unique solution.
The aim of this paper is to clarify and refine this result. H BM O is the set of locally integrable on R n functions such that k f k BMO = sup 1 | Q | Z Q | f − f Q | dx < ∞ , f Q = 1 | Q | Z Q f dx, where the supremum is taken over all cubes Q ⊂ R n with faces parallel to coordinatehyperplanes (or, alternatively, over all balls).It is well known that A ∈ BM O guarantees D ( A ) = W , (Ω) and [ u, u ] A = 0. Indeed,for u, v ∈ C ∞ (Ω) write Z Ω A ij u x j v x i dx = 12 Z Ω A ij ( u x j v x i − u x i v x j ) dx. The crucial fact is that u x i v x j − u x j v x i belongs to the Hardy space H ( R n ), and k u x i v x j − u x j v x i k H ( R n ) ≤ C k∇ u k L (Ω) k∇ v k L (Ω) . This fact can be proved using the commutator theorem from [17]. Much easier proof wasgiven later in [18]. There is a number of different equivalent definitions of H ( R n ), theproof of [18] used the following one. Let Φ be a smooth compactly supported functionwith R Φ dx = 1. Denote M Φ f ( x ) = sup t> | Φ t ∗ f | ( x ) , Φ t ( x ) = t − n Φ (cid:16) xt (cid:17) . H ( R n ) = (cid:8) f ∈ L ( R n ) : M Φ f ∈ L ( R n ) (cid:9) Since
BM O is dual to H ( R n ) [19], we arrive at Z Ω A ij u x j v x i dx ≤ C k A k BMO kk∇ u k L (Ω) k∇ v k L (Ω) , C = C ( n ) . Thus, the skew-symmetric bilinear form [ · , · ] A defined on C ∞ (Ω) × C ∞ (Ω) is continuoswith respect to both arguments in the norm of W , (Ω) and can be extended to the formon W , (Ω) × W , (Ω) satisfying | [ u, v ] A | ≤ C k∇ u k L (Ω) k∇ v k L (Ω) , [ u, v ] A = − [ v, u ] A for all u, v ∈ W , (Ω). Then the existence and uniqueness of a solution to (1) followsfrom the Lax-Milgram lemma.For other useful properties of BM O and Hardy spaces we refer the reader to [20] (seealso the excellent expository article [21]).A decade ago Maz’ya and Verbitsy [22] proved a reverse result. This result is formu-lated for a wide class of equations with lower-order terms. We cite here only the basicpart which relates to (1). Let L , ( R n ) be the closure of smooth finite functions withrespect to the norm k∇ u k L ( R n ) , and L − , ( R n ) be its dual. The operator − div ( A ∇ u ) : L , ( R n ) → L − , ( R n )is bounded if and only if A s = A + A T ∈ L ∞ ( R n ) , anddiv A c ∈ BM O − ( R n ) n , A c = A − A T . Here
BM O − ( R n ) denotes the set of distributions which can be represented as the diver-gence of a BM O vector field. So, there exists a matrix Φ with
BM O entries such thatdiv A c = div Φ. In the sense of generalized functions, for u, v ∈ C ∞ ( R n ) we have h− div ( A ∇ u ) , v i = h A s ∇ u, ∇ v i − h div A c ∇ u, v i = h A s ∇ u, ∇ v i − h div Φ ∇ u, v i = −h div (( A s + Φ) ∇ u ) , v i . This means that on smooth finite functions the operator is identical to an analogousoperator with symmetric part of the matrix bounded and skew-symmetric part from
BM O . The skew-symmetric part Φ can be found from Φ = −△ − curl div A c . Herethe divergence operator acts on a = a ij as div j a = ∂ x i a ij , and the curl of f = { f i } iscurl ij f = ∂ x j f i − ∂ x i f j . In dimension 2 the matrix A c itself belongs to BM O .The functions from
BM O are exponentially summable (the John-Nirenberg lemma[23]), and satisfy 1 | Q | Z Q | f − f Q | p dx ≤ ( Cp k f k BMO ) p , C = C ( n ) (16)7or any f ∈ BM O and cube Q ⊂ R n . Thus, for A ∈ BM O the limit in (15) is alwaysfinite, but need not be zero, as can be demonstrated by the example of log | x | .For A ∈ BM O
Zhikov proved the uniqueness of approximation solutions withoutusing the
BM O – H duality. In this case, it is sufficient to prove uniqueness for solutionscorresponding to the set of bounded right-hand sides, which is dense in W − , (Ω) (see[16] for details). If A ∈ BM O ∩ L ∞ (Ω), one can obtain the Meyers type estimate k∇ u k L q (Ω) ≤ C k f k L ∞ (Ω) for some q > C which depend only on k A k BMO and Ω. Since
BM O functions aresummable to any power, A ∇ u ∇ u ∈ L (Ω). By H¨older’s inequality | [ u, ϕ ] A | ≤ C k∇ u k L q (Ω) k A k L r (Ω) k∇ ϕ k L (Ω) , r − = 2 − − q − , for ϕ ∈ C ∞ (Ω). Approximating u by such ϕ we arrive at[ u, u ] A = Z Ω A ∇ u ∇ u = 0 , which implies uniqueness for approximation solutions corresponding to bounded right-hand sides.There is a variety results on equations of type (1) with A ∈ BM O (or equations oftype (3) with divergence-free a ∈ BM O − ). See, for instance, the survey article [24] onthe magnetogeostrophic equation and [25, 26] for results on regularity and qualitativetheory of solutions. Now we are ready to state the main result of this paper.
Theorem 4.1.
Let the matrix A satisfy the condition lim p →∞ p − k A k L p (Ω) < ∞ . (17) Then (1) has a unique solution.
By the John-Nirenberg estimate (16), matrices with
BM O elements satisfy (17). Itis easy to see that (17) is equivalent to the exponential summability of A : Z Ω exp( γ | A | ) dx = ∞ X p =0 γ p p ! Z Ω | A | p dx, (18)and by the Stirling formula γ p p ! Z Ω | A | p dx ∼ √ πp ( Leγ ) p , L = lim p →∞ p − k A k L p (Ω) . The series on the right-hand side of (18) converge if γ < ( Le ) − .8et M exp( γ | A | ) be the Hardy-Littlewood maximal function of exp( γ | A | ) ∈ L (Ω),which is continued by zero outside Ω. Clearly, | A | ≤ γ log M exp( γ | A | ) . By the result of Coifman and Rochberg [27], the right-hand side of the last expressionis in
BM O with the
BM O “norm” bounded by γ − C ( n ). So, (17) is equivalent to | A | having a BM O majorant.Let us note that the condition of exponential summability naturally arises in thetheory of qusiharmonic vector fields with unbounded distortion [28].It is easy to give an example of function satisfying (17) but not in
BM O . It followsfrom the definition of
BM O that for two touching cubes of the same size there holds | f Q − f Q | ≤ n +1 k f k BMO . (19)Let n = 2, x = ( x , x ). Take f ( x ) = log | x | if x x > f = 0 otherwise. Clearly, forsuch function (19) is not satisfied.The condition of theorem (4.1) is sufficient for the uniqueness but far from necessary.It is worth to note that the addition of a skew-symmetric matrix with zero divergenceto matrix A does not change the equation. Let C ∈ L (Ω) be a skew-symmetric matrixwith div C = 0, u ∈ W , (Ω) and ϕ ∈ C ∞ (Ω). We have h− div ( C ∇ u ) , ϕ i = Z Ω C ij u x j ϕ x i dx = Z Ω C ij ( uϕ ) x j dx − Z Ω uC ij ϕ x i x j dx = 0 . In dimension 2 this does not bring anything new since any skew-symmetric matrix 2 × (cid:18) c − c (cid:19) , c = const , and the addition of any bounded matrix to A does not affect (17). In dimension 3 thesituation is more interesting. Write the skew-symmetric matrix A as A = − a a a − a − a a , Aξ = a × ξ, a = ( a , a , a ) . The condition of zero divergence leads to ∇ × a = 0, which is satisfied by a = ∇ ϕ . Thiscan be also seen from (4). We can add any matrix of the form C ( ϕ ) = − ϕ x ϕ x ϕ x − ϕ x − ϕ x ϕ x , ϕ ∈ W , (Ω) , to A and the equation basically stays the same. This is the reason why in [22] the resultis given in terms of equivalence classes for n ≥ Lipschitz truncations. The proof of the main result
In this section we prove Theorem 4.1. The proof relies on the technique of Lipschitztrunctions. For the reader’s convenience we briefly remind the details. Let u ∈ W , (Ω)and g = M |∇ u | ), where M stands for the standard Hardy-Littlewood maximal function: M f ( x ) = sup 1 | B | Z B | f | dx, where the supremum is taken over all balls B ⊂ R n which contain x (uncentered maximalfunction) or are centered at x (centered maximal function). Then for almost all x, y ∈ Ωthere holds | u ( x ) − u ( y ) | ≤ C ( n ) | x − y | ( g ( x ) + g ( y )) , | u ( x ) | ≤ C dist ( x, ∂ Ω) g ( x ) . From these estimates it follows that on the set F ( λ ) = { g ≤ λ } ∪ ( R n \ Ω) the function u is Lipschitz with the Lipschitz constant Cλ . Using the McShane theorem [29], we canextend u | F ( λ ) to the whole space R n with the same Lipschitz constant Cλ . The resultingextension u λ is called the Lipschitz truncation of u . For further details on Lipschitztruncations and their applications we recommend [30].Let u be a solution to (1) with f = 0, i.e. Z Ω ( ∇ u + A ∇ u ) ∇ ϕ dx = 0 for all ϕ ∈ C ∞ (Ω) . (20)By approximation, one can take here Lipshitz ϕ vanishing on ∂ Ω.Take the test function ϕ = u λ in (2). Using the skew-symmetry of A we obtain Z { g ≤ λ } |∇ u | dx = − Z { g>λ } ( A + I ) ∇ u ∇ u λ dx ≤ Cλ Z { g>λ } ( | A | + 1) |∇ u | dx. Next, multiply this inequality by ελ − − ε , ε >
0, and integrate with respect to λ from 1to ∞ . Fubini’s theorem yields Z Ω |∇ u | (max(1 , g )) − ε dx ≤ Cε − ε Z Ω ( | A | + 1) |∇ u | ( g − ε − + dx. Using H¨older’s inequality and the boundedness of the maximal function in L , for small ε we obtain Z Ω |∇ u | (max(1 , g )) − ε dx ≤ Cε (cid:18)Z Ω |∇ u | dx (cid:19) / (cid:18)Z Ω g dx (cid:19) (1 − ε ) / (cid:18)Z Ω ( | A | + 1) /ε (cid:19) ε/ ≤ Cε (cid:18)Z Ω ( | A | + 1) /ε dx (cid:19) ε/ (cid:18)Z Ω |∇ u | dx (cid:19) − ε/ . Passing to the limit as ε → Z Ω |∇ u | dx ≤ C lim p →∞ k A k L p (Ω) p Z Ω |∇ u | dx, C = C ( n ) . u = 0 provided that the limit in (17) is small enough. The theorem is thusproved for A such that lim p →∞ p − k A k L p (Ω) < C ( n ) (21)for some positive constant C ( n ). Let A be a skew-symmetric matrix satisfying (17).Consider (1) with A replaced by ± tA with t > tA satisfies (21). Clearly, D ( ± tA ) = D ( A ) and [ u, u ] ± tA = ± t [ u, u ] A . For tA we have uniqueness, so [ u, u ] tA ≥ u ∈ D ( A ). Similarly, [ u, u ] − tA ≥ u ∈ D ( A ). Thus, [ u, u ] = 0 for all u ∈ D ( A ).This immediately implies the uniqueness of solutions and validity of (12). The proof ofTheorem 4.1 is complete. In this section we focus on problem (3) with “standard” solenoidal drift. A vector field a ∈ L (Ω) is called solenoidal (or divergence free) if div a = 0 in the sense of distributions,i.e. Z a ∇ ϕ dx = 0 for all ϕ ∈ C ∞ (Ω) . A solution to (3) is a function u ∈ W , (Ω) which satisfies Z Ω ( ∇ u + au ) ∇ ϕ dx = ( f, ϕ ) for all ϕ ∈ C ∞ (Ω) . (22)Using the same reasoning as above, one can show the existence of approximation solutionsif the solenoidal vector field a ∈ L n/ ( n +2) (Ω) for n ≥ , and a ∈ L log / L (Ω) for n = 2 . (23)In view of the embedding theorem this condition guarantees au ∈ L (Ω). Denote[ u, ϕ ] a = Z Ω au ∇ ϕ dx, u ∈ W , (Ω) , ϕ ∈ C ∞ (Ω) ,D ( a ) = { u ∈ W , (Ω) : | [ u, ϕ ] a | ≤ C k∇ ϕ k L (Ω) for all ϕ ∈ C ∞ (Ω) } . As above, the set D ( a ) coincides with the set of all solutions to (3), for a solution u theform [ u, ϕ ] a is extended to ϕ ∈ W , (Ω), (22) can be written in the form (9), substituting u as a test-function one obtains (11). Further on, if there is no ambiguity, we drop thesubscript a in the form [ u, ϕ ].The simplest condition (apart from the trivial a ∈ L ∞ (Ω)) which guarantees theexistence and uniqueness of a solution is a ∈ L n (Ω). For n >
2, by the Sobolev embeddingtheorem, (cid:12)(cid:12)(cid:12)(cid:12)Z Ω au ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k a k L n (Ω) k u k L (Ω) k∇ ϕ k L (Ω) , (24)so the form [ u, ϕ ] is continuous with respect to both arguments in the norm of W , (Ω) × W , (Ω), and the existence and uniqueness of a solution follows from the Lax-Milgramlemma. 11or n = 2, let Ω be a simply-connected domain. Since a = ( a , a ) is solenoidal, wecan find Q ∈ W , (Ω) such that a = − Q y , a = Q x . Rewrite (3) in the form (1): Z Ω u ( a ϕ x + a ϕ y ) dxdy = Z Ω Q ( u y ϕ x − u x ϕ y ) dxdy for all u ∈ W , (Ω) and ϕ ∈ C ∞ (Ω). Extend the function Q to the whole plane so that k Q k W , ( R n ) ≤ k Q k W , (Ω) . By the Poincar´e inequality, for any ball B ⊂ R there holds | B | − Z B | Q − ¯ Q | dx ≤ Z B | a | dx. Hence Q ∈ BM O and k Q k BMO ≤ C k a k L (Ω) . Using the duality of BM O and H , weobtain (24) for n = 2.A thorough study of regularity properties (boundedness, strong maximum principle,continuity, Harnack’s inequality) of solutions of second-order linear elliptic and parabolicequations with “rough” divergence free drifts from L n and Morrey spaces generalizing L n was done by Nazarov and Uraltseva in [31]. Interesting examples are due to Filonov [32].It is not hard to prove [2] that a ∈ L (Ω) guarantees the uniqueness of solutions andvalidity of the energy identity (12). Indeed, approximating u by smooth functions onecan prove that for a ∈ L (Ω) there holds Z Ω au ∇ ϕ dx = − Z Ω aϕ ∇ u dx, u ∈ W , (Ω) , ϕ ∈ C ∞ (Ω) , so (22) acquires the form Z Ω ∇ u ∇ ϕ dx = Z Ω aϕ ∇ u dx + ( f, ϕ ) for all ϕ ∈ C ∞ (Ω) . (25)Approximating T k ( u ) = max(min( u, k ) , − k ), k >
0, by bounded smooth functions, wecan set ϕ = T k ( u ) in (25), which gives Z Ω |∇ T k ( u ) | dx − ( f, T k ( u )) = Z Ω aT k ( u ) ∇ u dx = Z Ω a ∇ (cid:18)Z u T k ( s ) ds (cid:19) dx = 0 . Sending k to infinity, and using ∇ T k ( u ) = χ {| u | 1. A similar example for the problem −△ u + b ∇ u + div ( bu ) = f ∈ W − , (Ω), u ∈ W , (Ω) was constructed in [33], where the question of existence anduniqueness was studied in the framework of renormalized solutions.In [16] Zhikov proved the following result which improves the L (Ω) condition. Theorem (Zhikov) . If the solenoidal vector field a satisfies lim ε → ε k a k L − ε (Ω) = 0 , thenthe approximation solution of (3) is unique for each f ∈ W − , (Ω) . 12n dimension 2 this result can be strengthed. Theorem 6.1. Let n = 2 and the solenoidal vector field a = ( a , a ) satisfy lim ε → √ ε k a k L − ε (Ω) = 0 . (26) Then for any f ∈ W − , (Ω) equation (3) has a unique solution. We shall obtain this theorem as a partial case of a more general statement.Recall that the Morrey space M p (Ω), 1 ≤ p ≤ ∞ , is the set of all integrable functionssuch that k f k M p (Ω) := sup R − n (1 − /p ) Z Ω ∩ B R | f | dx < ∞ , where the supremum is taken over all balls B R of radius R . It is well known that for f ∈ M n (Ω) the Riesz potential I Ω f ( x ) = Z Ω | x − y | − n | f ( y ) | dy is exponentially summable and satisfies [34, proof of Lemma 7.20] Z Ω | I Ω f | q dx ≤ n ( n − q − ω n q q (diam Ω) n k f k M n (Ω) , (27)where positive constants c and c depend only on n , p . We shall also use the followingsimple potential estimate [34, Lemma 7.12]. Let 1 ≤ q ≤ ∞ , 0 ≤ δ = p − − q − < µ , µ = n − . Then k I Ω f k L q (Ω) ≤ (cid:18) − δµ − δ (cid:19) − δ ω − µn | Ω | µ − δ k f k L p (Ω) . (28) Theorem 6.2. Let a ∈ M n (Ω) and satisfy (23) . Then (3) has a unique solution. Thesame conclusion also holds if lim ε → ε /n k a k L n − ε (Ω) < ∞ . (29) Proof. Here we use the same notation as in the proof of Theorem 4.1. Let u be a solutionto (3) with f = 0 and u λ be the Lipschitz truncation of u . Using u λ as a test function in(22), we obtain Z { g ≤ λ } |∇ u | dx = − Z Ω au ∇ u λ dx − Z { g>λ } ∇ u ∇ u λ dx = Z { g>λ } ( a ( u λ − u ) − ∇ u ) ∇ u λ dx ≤ Cλ Z { g>λ } ( | a | · | u − u λ | + |∇ u | ) dx. (30)For almost all x ∈ Ω by the Poincar´e inequality [34, Lemma 7.16] there holds | u − u λ | ( x ) ≤ (diam Ω) n |{ g ≤ λ } ∩ Ω | Z { g>λ } |∇ ( u − u λ ) || x − y | n − dy. (31)13et λ be such that |{ g ≤ λ } ∩ Ω | > δ (diam Ω) n , δ a small positive number. Let V ( x ) = R Ω | x − y | − n | a ( x ) | dx . From (30), (31), for λ ≥ λ we have Z { g ≤ λ } |∇ u | dx ≤ Cλ Z { g>λ } |∇ u | dx ++ Cλ Z { x ∈ Ω: g ( x ) >λ } Z { y ∈ Ω: g ( y ) >λ } | a ( x ) | · | x − y | − n · ( |∇ u ( y ) | + λ ) dx dy ≤ Cλ Z { g>λ } |∇ u | dx + Z { g>λ }∩ Ω ( |∇ u | + λ ) V dx. Multiply this relation by ελ − − ε , 0 < ε < / 2, and integrate with respect to λ from λ to + ∞ . Fubini’s theorem yields Z Ω |∇ u | (max( λ , g )) − ε dx ≤ Cε Z Ω ( |∇ u | g − ε + g − ε ) V dx + Cε Z Ω g − ε |∇ u | dx. By H¨older’s inequality, Z Ω |∇ u | ( λ + g ) − ε dx ≤ Cε "(cid:18)Z Ω g dx (cid:19) − ε (cid:18)Z Ω V /ε dx (cid:19) ε/ + Z Ω g − ε |∇ u | dx . (32)If a ∈ M n (Ω), from (27) we obtain (cid:18)Z Ω V ( x ) /ε dx (cid:19) ε/ ≤ C (diam Ω) nε/ k a k M n (Ω) ε/ . If a satisfies (29), then using (28) with p = 2 n/ (2 + ε ) and q = 2 /ε we have (cid:18)Z Ω V ( x ) /ε dx (cid:19) ε/ ≤ Cε (1 − n ) /n | Ω | ε ( n − / n k a k L n − ε (Ω) . Substituting this estimate into (32), using the boundedness of the Hardy-Littlewoodmaximal function in L and sending ε → 0, we arrive at Z Ω |∇ u | dx ≤ CJ Z Ω g dx ≤ CJ Z Ω |∇ u | dx where J is either k a k M n (Ω) or lim ε → ε / k a k L n − ε (Ω) . Hence, R Ω |∇ u | dx = 0 provided that J is sufficiently small. This assumption can be removed by the same argument as in theproof of Theorem 4.1.Theorem 6.2 can be obtained as a corollary of Theorem 4.1 if we find a suitablerepresentation of a solution to div A = a in terms of integral potentials with kernels K ( x, y ) = O ( | x − y | − n ). In this case applying (27) (or (28)) we would obtain (17) for A .In dimension 2 this is easy. Also this is simple provided that the normal component of a on ∂ Ω is zero, in which case a solution is given by (5). In the general case, this is alsopossible, but requires certain analytical work. Let n = 3 and a be a smooth solenoidalvector field. Let Ω be star-shaped with respect to a ball B ⊂ Ω. For y ∈ B and x ∈ Ω the14unction w ( x, y ) = R ∇ a ( y + t ( x − y )) × t ( x − y ) dt satisfies rot x w = a . Let ψ ∈ C ∞ ( B ), R Ω ψ dx = 1. The function W ( x ) = R B ψ ( y ) w ( x, y ) dy solves rot W = a . Interchangingthe order of integration, we arrive at W ( x ) = Z Ω a ( y ) × x − y | x − y | Z + ∞| x − y | (cid:18) − | x − y | r (cid:19) r ψ (cid:18) x + r y − x | y − x | (cid:19) dr dy. There remains the task of checking the validity of this formula (say, in the spirit of [13]),and for domains of more complex geometry this is not directly applicable. In the proofof Theorem 6.2 we circumvent these problems.Condition (29) means that a is from the grand Lebesgue space L n ) (Ω) introduced byIwaniec and Sbordone [35], which is the set of functions f integrable to any power lessthan n with the finite norm k f k L n ) (Ω) = sup ≤ s 1. Further account of properties of grand Lebesgue spacesand their investigation by methods of interpolation theory can be found in [36]. Theclosure of L n (Ω) in L n ) (Ω) is strictly less than the latter space and is characterized bylim sup ε → ε /n k f k L n − ε (Ω) = 0.For a solenoidal vector field a ∈ L (Ω) one can easily construct an approximationsolution of (3) for bounded right-hand sides, f ∈ L ∞ (Ω) (or, say, f = f i,x i + g , g ∈ L q/ (Ω), f i ∈ L q (Ω), q > n ). This fact follows from the supremum estimate k u k L ∞ (Ω) ≤ C k f k L ∞ (Ω) which is valid for bounded solenoidal a with the constant C independentof a . Applying the same reasoning as in Theorem 6.2, one can prove the uniquenessof approximation solution of (3) with a from the Morrey space M n (Ω) and the right-hand side f from L ∞ (Ω) without requiring (23). Now, let f be an arbitrary element of W − , (Ω). It can be approximated by bounded f j . Let u j be approximation solutions of(3) corresponding to f j . Since in this case approximation solutions are uniquely defined,the difference of any two approximation solutions is also an approximation solution,satisfying k u j − u k k W , (Ω) ≤ C k f j − f k k W − , (Ω) . Therefore, the sequence u j has a stronglimit u , which does not depend on the choice of approximation of f . It would be naturalto call this limit a solution to (3) corresponding to the right-hand side f . The limitfunction can be unbounded, so the term au need not be integrable here. The questionis how to understand the equation. For instance, using the Sobolev representation andFubini’s theorem, for bounded u we can transform the drift term in the integral identityas follows: Z Ω au ∇ ϕ dx = ( nω n ) − Z Ω ∇ u ( y ) dy Z Ω ( x − y ) a ∇ ϕ ( x ) | x − y | n dx, which is well defined ( a ∇ ϕ ∈ M n (Ω) if ∇ ϕ is bounded) and allows the passage to thelimit with respect to the convergence of ∇ u in L (Ω). Acknowledgements : The work was partially supported by the Russian Foundationfor Basic Research, project № № eferences [1] Fannjiang MA, Papanicolaou GC. Diffusion in turbulence. Probab Theory Related Fields.1996; 105: 279–334.[2] Zhikov VV. Diffusion in an incompressible random flow. Funct Anal Appl. 1997; 31(3):156–166.[3] Osada H. Diffusion processes with generators of generalized divergence form. J. Math.Kyoto Univ. 1987; 27 (4) : 597–619.[4] Spivak M. Calculus on manifolds: A modern approach to classical theorems of advancedcalculus. Reading(MA): Addison-Wesley, 1965.[5] Morrey Ch B. Multiple integrals in the calculus of variations. Berlin: Springer, 1966.[6] Mitrea D, Mitrea M, Shaw MC. Traces of differential forms on Lipschitz domains, theboundary De Rham complex, and Hodge decompositions. Indiana Univ Math J. 2008;57(5): 2061–2095.[7] Kress R. Die Behandlung zweier Randwertprobleme f¨ur die vektorielle Poissongleichungnach einer Integralgleichungsmethode. Arch Rational Mech Anal. 1970; 39(3): 206–226.[8] Mitrea D, Mitrea M, Pipher J. Vector potential theory on nonsmooth domains in R andapplications to electromagnetic scattering. J Fourier Anal Appl. 1997; 3(2): 131–192.[9] Dubinskii YA. Some coercive problems for the system of Poisson equations. Russian J MathPhys. 2013: 20(4): 402–412.[10] Dubinskii YA. On some boundary value problems and vector potentials of solenoidal fields.J Math Sci. 2014; 196(4): 524–534. (translated from Problemy Matematicheskogo Analiza2013; 74: 75–84)[11] Ladyzhenskaya OA, Solonnikov VA. Some problems of vector analysis and generalizedformulations of boundary-value problems for the NavierStokes equations. J Soviet Math.1978; 10(2): 257–286. (translated from Zap Nauchn Sem LOMI 1976; 59: 81–116)[12] Ladyzhenskaya OA. Relationship between the Stokes problem and decompositions of thespaces ˚ W and W ( − . St Petersburg Math J. 2002; 13(4): 601–612. ( translated fromAlgebra i Analiz. 2001; 13(4): 119–133)[13] Ladyzhenskaya OA. On a construction of basises in spaces of solenoidal vector-valued fields.J Math Sci (N. Y.) 2005. 130(4): 4827–4835. (translated from Zap Nauchn Sem POMI.2003; 306: 92–106)[14] Kapitanski LV, Pileckas KI. On some problems of vector analysis. Zap Nauchn Sem LOMI.1984; 138: 65–85.[15] Kato T, Mitrea M, Ponce G, Taylor M. Extension and representation of divergence-freevector fields on bounded domains. Math Research Letters. 2000; 7: 643–650.[16] Zhikov VV. Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms. Funct Anal Appl. 2004;38(3): 173–183. 17] Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in severalvariables. Ann Math. 1976; 103: 611–635.[18] Coifman R, Lions PL, Meyer Y, Semmes S. Compensated compactness and Hardy spaces.J Math Pures Appl. 1993; 72: 247–286.[19] Fefferman C, Stein EM. H p spaces of several variables. Acta Math. 1972; 129: 137–193.[20] Stein EM. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory in-tegrals. Princeton (NJ): Princeton Univ. Press; 1993.[21] Semmes S. A primer on Hardy spaces, and some remarks on a theorem of Evans and M¨uller.Comm Partial Differential Equations. 1994; 19(1&2) : 277–319.[22] Mazya VG, Verbitskiy IE. Form boundedness of the general second-order differential op-erator. Comm Pure Appl Math. 2006; 59(9): 1286–1329.[23] John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math.1961; 14: 415–426.[24] Friedlander S, Rusin W, Vicol V. The magneto-geostrophic equations: a survey. Proceed-ings of the St. Petersburg Mathematical Society, Volume XV: Advances in MathematicalAnalysis of Partial Differential Equations. 2014. D. Apushkinskaya and A.I. Nazarov, eds.pp. 53–78[25] Friedlander S, Vicol V. Global well-posedness for an advective-diffusion equation arising inmagneto-geostrophic dynamics. Ann Inst H Poincar´e Anal Non Lin´eaire. 2011; 28: 283–301.[26] Serigin G, Silvestre L, ˇSver´ak V, Zlatoˇs A. On divergence-free drifts. J Differential Equa-tions. 2012; 252: 505–540.[27] Coifman R, Rochberg R. Another characterization of BMO. Proc Amer Math Soc. 1980;79(2): 249–254.[28] Iwaniec T, Sbordone C. Quasiharmonic fields. Ann Inst H Poincar´e Anal Non Lin´eaire.2001; 18(5): 519–572.[29] McShane EJ. Extension of range of functions. Bull Amer Math Soc. 1934; 40: 837–842.[30] Diening L, M´alek J, Steinhauer S. On Lipschitz truncations of Sobolev functions (withvariable exponent) and their selected applications. ESAIM: COCV. 2008; 14(2): 211–232.[31] Nazarov AI, Uraltseva NN. The Harnack inequality and related properties for solutions toelliptic and parabolic equations with divergence-free lower-order coefficients. St PetersburgMath J. 2012; 23(1): 93–115. (translated from Algebra i Analiz. 2011: 23(1): 136–168)[32] Filonov N. On the regularity of solutions to the equation −△ u + b ∇ u = 0. J Math Sci. (N.Y.) 2013; 195(1): 98–108. (translated from Zap. Nauchn. Sem. POMI. 2013; 410: 168–186)[33] Briane M, Casado-D´ıaz J. A class of second-order linear elliptic equations with drift: renor-malized solutions, uniqueness and homogenization. J Potential Anal. 2015; 43(3): 399–413.[34] Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Reprint ofthe 1998 ed. Berlin: Springer, 2001. 35] Iwaniec T, Sbordone C. On the integrability of the Jacobian under minimal hypotheses.Arch. Rational Mech. Anal. 1992; (119): 129–143.[36] Fiorenza A, Karadzhov GE. Grand and small Lebesgue spaces and their analogs. Journalfor Analysis and its Applications. 2004; 23(4): 657–681.35] Iwaniec T, Sbordone C. On the integrability of the Jacobian under minimal hypotheses.Arch. Rational Mech. Anal. 1992; (119): 129–143.[36] Fiorenza A, Karadzhov GE. Grand and small Lebesgue spaces and their analogs. Journalfor Analysis and its Applications. 2004; 23(4): 657–681.