LORENTZ ESTIMATES FOR WEAK SOLUTIONS OF QUASI-LINEAR PARABOLIC EQUATIONS WITH SINGULAR DIVERGENCE-FREE DRIFTS TUOC PHAN

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1 LORENTZ ESTIMATES FOR WEAK SOLUTIONS OF QUASI-LINEAR PARABOLIC EQUATIONS WITH SINGULAR DIVERGENCE-FREE DRIFTS TUOC PHAN Abstract This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts In this class of equations, the principal terms are vector field functions which are measurable in (x, t)-variable, and nonlinearly dependent on both unknown solutions and their gradients Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the Jonh-Nirenberg space The results are even new when the drifts are identically zero because they do not require solutions to be bounded as in the available literatures In the linear setting, the results of the paper also improve the standard Calderón-Zygmund regularity theory to the critical borderline case When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known, available results The approach is based on the perturbation technique introduced by Caffarelli-Peral together with a double-scaling parameter technique, and the maximal function free approach introduced by Acerbi- Mingione Introduction This paper establishes local interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions of the following class of quasi-linear parabolic equations with singular divergence-free drifts, and with conformal boundary condition () u t div [ A(x, t, u, u) b(x, t)u F(x, t) ] = f (x, t) (x, t) Ω (, T), A(x, t, u, u) b(x, t)u F(x, t), ν = (x, t) Ω (, T), u(x, ) = u (x), x Ω where Ω is a bounded domain in R n with boundary Ω, ν is the unit outward normal vector on Ω, f : Ω (, T) R is a given measurable function, F, b : Ω (, T) R n are given vector field functions, and u is an unknown solution with a given initial condition u for which we do not require any regularity Moreover, T is a given fixed positive number, and the principal term A = A(x, t, s, ξ) : Ω (, T) K R n R n is a given vector field We assume that A(,, s, ξ) is measurable in Ω T = Ω (, T) for every (s, ξ) K R n ; A(x, t,, ξ) Hölder continuous in K for ae (x, t) Ω T and for all ξ R n ; and A(x, t, s, ) differentiable in R n for each s K and for ae (x, t) Ω T Here, K is an open interval in R, which could be the same as R We assume in addition that there exist constants Λ > and α (, ] such that A satisfies the following natural growth conditions () (3) (4) A(x, t, s, η) A(x, t, s, ξ), η ξ Λ η ξ, for ae (x, t) Ω T, s K, ξ, η R n, A(x, t, s, ξ) + ξ ξ A(x, t, s, ξ) Λ ξ, for ae (x, t) Ω T, s K, ξ R n, A(x, t, s, ξ) A(x, t, s, ξ) Λ ξ s s α s, s K, for ae (x, t) Ω T, ξ R n Under the conditions () (4) and with F = b =, the class of equations () contains the well-known quasi-linear parabolic equations with zero-flux boundary condition If F =, but b, the equation () is the standard nonlinear advection-diffusion equations The drift term b considered in this paper could be singular Due to its relevance in many applications such as in fluid dynamics and mathematical biology Date: January 9, 8

2 T PHAN (see [4,, 7, 43, 44, 47] for examples), we are particularly interested in the case that b is divergence-free, ie (5) div [b(, t)] =, in the sense of distributions in Ω, for ae t (, T) On one hand, when b = and F, f are sufficiently regular, the C,α -regularity theory for bounded, weak solutions of this class of equations () has been investigated extensively in the classical work, see for example [,3,3,3,45], assuming some regularity of A in (x, t, s, ξ) Ω T K R n On the other hand, when b, F, f are not so regular or when A is discontinuous in (x, t), one does not expect those mentioned Schauder s type estimates for weak solutions of () to hold It is therefore mathematically interesting, and essentially important to search for regularity estimates of Calderón-Zygmund type for gradients of weak solutions in Lebesgue spaces In particular, in these situations, this kind of Calderón-Zygmund regularity estimates is vital in studying many questions in nonlinear equations and systems of equations, see [7] for example In this perspective, it is known that to establish the Calderón-Zygmund theory, the class of considered equations must be invariant under the scalings and dilations, see [46] for more geometric intuition of this issue However, due to the fact that the nonlinearity of the principal term A depends on u as its variable, the class of this equations () is not invariant under the scalings and dilations (6) u u/λ, and u(x, t) u(rx, r t), for all positive numbers r, λ r Due the lack of this homogeneity, Calderón-Zygmund type regularity theory for weak solutions of () becomes delicate, and is still not completely understood In a simpler case when A is independent on the variable s K, and b = f =, the equation () is reduced to (7) u t div [A(x, t, u)] = div [F] in Ω T, and the W,q -regularity estimate for weak solutions of equations (7) has been non-trivially and extensively developed by many authors for both elliptic, parabolic settings and also for p-laplacian type equations, for example see [6 8,,, 4, 7, 8, 3, 34, 37] In the recent work [7, 38, 39], the W,q -regularity estimates for weak solutions of equations () with b = is addressed, and the W,q -regularity estimates are established for bounded weak solutions To overcome the loss of homogeneity that we mentioned, in [7, 38, 39], we introduced some double-scaling parameter technique Essentially, we study an enlarged class of double-scaling parameter equations of the type () Then, by some compactness argument, we successfully applied the perturbation method in [] to tackle the problem Careful analysis is required to ensure that all intermediate steps in the perturbation process are uniform with respect to the scaling parameters See also a very recent work [9] for further implementation of this idea for which global regularity theory for bounded weak solutions of some class of degenerate elliptic equations is obtained In all mentioned papers [9, 7, 38, 39], the boundedness assumption on the solutions is essential to start the investigation of W,q -theory This is because the approach uses maximum principle for the unperturbed equations to implement the perturbation technique We would like to refer also to [5] for which the W,q -theory for parabolic p-laplacian type equations of the form () is also achieved, but only for continuous weak solutions plus other assumptions on A In this paper, we establish regularity estimates in Lorentz spaces for gradients of weak solutions of () by assuming that the solutions are in the BMO space, ie the critical borderline case, and including the singular drifts b We achieve this in Theorem and Theorem, Theorem 3 below Our paper therefore generalizes the results in [5, 9, 7, 39] for () by relaxing the boundedness assumption on solutions, and putting into the context of Lorentz space setting Even in the linear case, and with f =, our results are also stronger than the classical Calderón-Zygmund results Precisely, in this case, () is reduced to (8) u t div [A (x, t) u] = div [bu + F] and the classical Calderón-Zygmund theory gives u L p C[ F L p + u L b L p]

3 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 3 Our results in Theorem, Theorem, Theorem 3 below improve this estimate by replacing u L by its borderline case [[u]] BMO See also [43] for some similar results in this direction for linear equations and with more regularity assumptions on b At this point, we also would like to note that when b, F, f satisfies some certain regularity conditions, weak solutions of (8) are proved in [44, 47] to be in C α, with some α (, ) The results in this paper therefore can be considered as the Sobolev counter part of this result, but for more general nonlinear equations Unlike [9, 7, 38, 39] which use double-scaling parameter, we only use single scaling parameter in the class of our equations (see [4, 4]) Precisely, we will investigate the following class of equations (9) u t div [ A(x, t, λu, u) b(x, t)u F(x, t) ] = f (x, t), in Ω T, A(x, t, λu, u) b(x, t)u F(x, t), ν =, on Ω (, T), u(, ) = u ( ), in Ω, with the scaling parameter λ As we will see in Subsection, this class of equations is the smallest one that is invariant with respect to the scalings and dilations (6), and that contains the class of equations () When λ =, f =, and b =, the equation (9) clearly becomes the equation (7) This paper therefore recovers all known results for (7) such as [8, 7, 43] In this paper, B R (y) denotes the ball in R n with radius R > and centered at y R n If y =, we write B R = B R () Also, for each z = (x, t ) R n+, we write When z =, we write Q R (z ) = B R (x ) Γ R (t ), with Γ R (t ) = (t R, t + R ) Q R = Q R (, ), Γ R = Γ R () For a measurable set U R n+, for some ρ >, and for a locally integrable f : U R n, the bounded mean oscillation semi-norm of f is defined by [[ f ]] BMO(U,ρ ) = sup f (x, t) f Qρ (z z =(x,t ) U Q ρ (z ) U ) U dxdt, where Q ρ (z ) U <ρ<ρ f Qρ (z ) U = Q ρ (z ) U Q ρ (z ) U f (x, t)dxdt For each p > and q (, ], the Lorentz quasi-norm of f on U is defined by { p s () f L p,q (U) = q { } } (x, t) U : f (x, t) > s q/p /q ds s, if q <, sup s> s { } (x, t) U : f (x, t) > s /p, if q = The set of all measurable functions f defined on U so that f L p,q (U) < is denoted by L p,q (U) and called Lorentz space with indices p and q It is clear that L p,p (U) = L p (U) - the usual Lebesgue space Moreover, L p,q (U) L p,r for all p > and < q < r When q =, the space L p, (U) is usually called weak-l p (U) space or Lorentz-Marcinkiewicz space See [4, Chapter 4], for example, for more details on Lorentz spaces Our first main result is the interior regularity estimates for the gradients of solutions of () Theorem Let Λ >, M >, p >, q (, ], and α (, ] Then, there exists a sufficiently small constant δ = δ(p, q, n, Λ, M, α ) > such that the following statement holds For every R >, let A : Q R K R n R n be a Carathéodory map satisfying ()-(4) on Q R K R n for some R > and some open interval K R, and () [A] BMO(QR,R) := sup z =(x,t ) Q R, <ρ R [ Q ρ (z ) Q ρ (z ) sup ξ R n \{}, s K A(x, t, s, ξ) Ā Bρ (x )(t, s, ξ) ] dxdt δ ξ

4 4 T PHAN Then, if F L p,q (Q R, R n ), f L n p,n q (Q R ), and u is a weak solution of u t div[a(x,, t, λu, u) bu F] = f (x, t) in Q R, with [[λu]] BMO(QR,R) M for some λ, and [[u]] BMO(QR,R)b L p,q (Q R, R n ) for some given divergencefree vector field b defined on Q R, there holds [ u L p,q (Q R ) C F L p,q (Q R ) + R Q R p pn f L n p,n q (Q R ) () + [[u]]bmo(qr,r)b ] L p,q (Q R ) + Q R p u L (Q R ), where n = n+ n+4, and C is a constant depending only on q, p, n, Λ, α, M, K Local regularity estimates near the boundary are not only interesting by themselves, but also important in many problems because they only require local information on data Our next result is the local regularity estimate on the boundary Ω for weak solutions u of () In this theorem, for z = (y, t) Ω R, and R >, we write When z = (, ), we write Ω R (y) = Ω B R (y), K R (z) = Ω R (y) Γ R (t), T R (z ) = ( Ω B R (y)) Γ R (t) Ω R = Ω R (), K R = K R (, ), T R = T R (, ) For each ˆx Ω, we assume that div[b] = in Ω R ( ˆx) and b, ν = on B R ( ˆx) Ω in the sense that (3) b(x, t), ϕ(x) dx =, ϕ C (B R( ˆx)), for ae t (, T) Ω R ( ˆx) Theorem Let M >, Λ >, p >, q (, ], and α (, ] Then, there exists a sufficiently small constant δ = δ(p, q, n, Λ, M, α ) > such that the following statement holds true Suppose that Ω and for some R >, Ω B R is C, and suppose that A : K R K R n R n is a Carathéodory map satisfying ()-(4) on K R K R n for some R > and some open interval K R, and (4) [A] BMO(KR,R) := sup z =(x,t ) K R, K ρ (z ) <ρ R K ρ (z ) [ sup ξ R n \{}, s K A(x, t, s, ξ) Ā Ωρ (x )(t, s, ξ) ] dxdt δ ξ Then, for every F L p,q (K R, R n ), f L n p,n q (K R ), if u is a weak solution of { ut div[a(x, t, λu, u) bu F] = f (x, t), in K (5) R, A(x, t, λu, u) bu F, ν =, on T R, satisfying [[λu]] BMO(KR,R) M, and [[u]] BMO(KR,R)b L p,q (K R, R n ) with some λ and some given divergence-free vector field b defined on K R and satisfying (3) at ˆx =, there holds [ u L p,q (K R ) C F L p,q (K R ) + R K R p pn f L n p,n q (K R ) (6) + [[u]]bmo(kr,r)b ] L p,q (K R ) + K R p u L (K R ), where n = n+ n+4, and C is a constant depending only on q, p, n, Λ, α, M, K Theorem, Theorem 3 are still valid if we replace Q ρ (z ) by ˆQ ρ (z ) = B ρ (x ) (t r, t ] and K ρ (z ) by ˆK ρ (z ) = Ω ρ (x ) (t ρ, t ] As a consequence, the following global regularity estimates in Lorentz space for gradients of weak solutions of (9) can be obtained Theorem 3 Let M >, Λ >, p >, q (, ], and α (, ] Then, there exists a sufficiently small constant δ = δ(p, q, n, Λ, M, α ) > such that the following statement holds true Suppose that Ω is C,

5 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 5 and suppose that A : Ω T K R n R n is a Carathéodory map satisfying ()-(4) on Ω T R R n for some T > and some open interval K R, and sup z =(x,t ) Ω ( t,t), <ρ r ˆQ ρ (z ) (Ω ( t, T)) ˆQ ρ (z ) Ω T [ sup ξ R n \{}, s K A(x, t, s, ξ) Ā Ωρ (x )(t, s, ξ) ] dxdt δ, ξ for some r >, t (, T) Then, for every F L p,q (Ω T, R n ), f L n p,n q (Ω T ), if u is a weak solution of (9) satisfying [[λu]] BMO(ΩT,r) M and [[u]] BMO(ΩT,r)b L p,q (Ω T, R n ) with some λ and some given vector field b satisfying (3) at every ˆx Ω, there holds u L p,q (Ω ( t,t)) C [ F L p,q (Ω T ) + f L n p,n q (Ω T ) + [[u]]bmo(ωt,r)b L p,q (Ω T )], where n = n+ n+4, and C is a constant depending only on q, p, n, Λ, α, M, K, r, Ω, t, T Several remarks are worth mentioning regarding Theorem, Theorem, and Theorem 3 Firstly, we reinforce that the most important improvement in Theorem, Theorem, and Theorem 3 is that they relax and do not requires the solutions to be bounded as in the known work [5, 9, 7, 38, 39] This is completely new even for the case b = and f =, in comparison to the known work that we already mentioned for both the Schauder s regularity theory and the Sobolev s one regarding weak solutions of equations () To overcome the loss of boundedness from the assumption, instead of applying maximum principle during the approximation process, we directly derive and carefully use some delicate analysis estimates and Hölder s regularity estimates for solutions of the corresponding homogeneous equations, see the estimates (35) and (38) for examples These estimates are first observed in the work [4, 4] but for elliptic equations In a related context, interested readers may see [5, 33] for other study on C α -regularity of weak, BMO solutions Secondly, we also note that due to the availability of f, which is scaled differently compared to F and u, the approach based on Hardy-Littlewood maximal function, and harmonic analysis used in [7, 8,, 4, 4, 46] does not seem to produce our desired estimates here Instead, we use the maximal-function free approach introduced in [], and also used in [,5,6] This paper seems to be the first one that treat the equations () with in-homogeneuous f in the Lorentz space setting In addition, this paper also treats quasi-linear equations with non-homogeneous singular drifts b, which has not done before As one will find in the proof, to deal with b, we introduce the function G(x, t) [[u]] BMO b(x, t) which has the same scaling properties as F, u This key fact plays an essential role in the proof Thirdly, we note that when λ =, f =, and b =, Theorem, Theorem and Theorem 3 recover and sharpen results in [6 8,,4,7,8,7,3,34,37,43] when restricting to the class of equations () in which A is independent on u K, see Remark 4 for more details on this See also [6, 9, 4] for some other related work with more regular f, F This paper therefore not only unifies both W,q -theories for () and (7) but also extends the theory to the Lorentz regularity estimate setting Lastly, observe that all papers such as [6, 7, 9, 37], to cite a few, regarding the W,q -regularity estimates in non-smooth domains only establish globally regularity estimates Our paper provides the regularity estimates locally for both the interior and the boundary one Our Theorem, Theorem can be considered as some high regularity estimates of Caccioppoli type which are important for many practical purposes for which local information is available and required Certainly, our local regularity estimates imply the global ones as Theorem 3 However, it is generally impossible to derive local estimates directly from the global ones in [6 9, 37] Remark 4 Two important points are worth pointing out (i) This paper does not require any regularity assumption on the initial data u in (), compared to [6, 8, 37] in which it is assumed that u = Moreover, M is not required to be small Note also that the condition [[u]] BMO b L p,q is trivial if b = Similarly, the condition [[λu]] BMO M is always satisfied when λ =

6 6 T PHAN (ii) If λ =, b = and f =, known results for (7) such as [5, 6, 8, 37] provide the estimates of the form (7) u L p C [ F L p + ] Our estimates in Theorems -3 are invariant under the scalings and dilations, and they do not contains the inhomogeneous constant, ie the number in the right hand side of (7) Our results are natural, and sharp We now conclude this section by outlining the organization of this paper Section reviews some definitions, and proves preliminaries needed in the paper Perturbation arguments, and approximation estimates are given in Section 3 Section 4 establishes estimates of level sets of gradients of solutions The proofs of main theorems, Theorems -3 are given in Section 5 The paper concludes with an appendix, Appendix A, giving proofs for some reverse Hölder s inequalities needed in the paper Definitions, and preliminaries Invariant properties, and definitions of weak solutions Let λ, and Q R R n+ be the parabolic cylinder of radius R Let us consider a weak solution u of u t div [A(x, t, λ u, u) bu(x, t) F(x, t)] = f (x, t) in Q R Then it is simple to check that for some fixed λ >, the rescaled function () v(x, t) = is a weak solution of u(x, t) λ for (x, t) Q R, v t div [ Â(x, ˆλv, v) b(x, t)v(x, t) ˆF(x, t) ] = ˆ f in Q R for ˆλ = λλ,  : Q R K R n R n defined by () Â(x, t, s, ξ) = Moreover, let λ = Rλ, (3) A(x, t, s, λξ), and f ˆ(x, t) = λ f (x, t), ˆF(x, t) = λ F(x, t), (x, t) Q R λ ṽ(x, t) = u(rx, R t), Ã(x, t, s, ξ) = A(Rx, R t, s, ξ) (x, t) Q, s K, ξ R n, and R F(x, t) = F(Rx, R t), f (x, t) = R f (Rx, R t), b(x, t) = Rb(Rx, R t), (x, t) Q Then, ṽ is a weak solution of ṽ t div [Ã(x, t, λṽ, ṽ) bṽ F] = f, in Q This is the main reason that we study the class of equation (9) with a parameter λ, instead of () Remark It is not too hard to see that if A : Q R K R n R n satisfies conditions () (4) on Q R K R n, then the rescaled vector field  : Q R K R n R n defined in () also satisfies the conditions () (4) on Q R K R n R n with the same constants Λ, α The same conclusion also holds for à : Q K R n R n defined in (3) Moreover, [Â] BMO(QR,R) = [Ã] BMO(Q,) = [A] BMO(QR,R), and [[ˆλv]] BMO(QR,R) = [[ λṽ]] BMO(Q,) = [[λ u]] BMO(QR,R) With respect to the scalings and dilations, the following remark follows directly from (), see also [4, Remark 47]

7 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 7 Remark For all < p, r < and for all < q, if f is a measurable function defined on a measurable set U R n+, then f r L p,q(u) = f r L rp,rq (U) Moreover, for a measurable function f defined on Q R with some R >, then f L p,q (Q ) = R (n+)/p f L p,q (Q R ), where f (x, t) = f (Rx, R t), (x, t) Q Let us now give the precise definition of weak solutions that is used throughout the paper Definition 3 Let K R be an interval, let Λ >, α (, ] and α > Also, let Ω R n be open, and bounded domain with boundary Ω, and A : Ω T K R n R n satisfy conditions () (4) on Ω T For each F L (Ω T ; R n ), f L (n+) n+4 (Ω T ) and λ, a function u is called weak solution of u t div [ A(x, t, λu, u) bu F ] = f (x, t), in Ω T, A(x, t, λu, u) bu F, ν =, on Ω (, T), if λu(x, t) K for ae (x, t) Ω T, u L ((, T), L (Ω)) L ((, T), W, (Ω)), [[u]] BMO(ΩT )b L α loc (Ω T, R n ), and for all ϕ C (Ω T ) with ϕ(, ) = ϕ(, T) = uϕ t dxdt + A(x, t, λu, u) bu F, ϕ dxdt = f (x, t)ϕ(x, t)dxdt Ω T Ω T Ω T Here, L p (U, R n ) for p < is the Lebesgue space consists all measurable functions f : U R n such that f p is integrable on U, and W,p (U) is the standard Sobolev space on U Moreover,, is the Euclidean inner product in R n Remark 4 When b, we require that the solution u BMO(Ω T ) to insure that bu, ϕ dz is Ω T well-defined for a singular vector field b Indeed, for b L α loc (Ω T ) with some α >, if ϕ C (Q) with some cube Q Ω T, since div [b(, t)] =, we can write bu, ϕ dz = b(u ū Q ), ϕ dz Ω T Q Then, it follows from the Hölder s inequality that ) /α ( ) α ( bu, ϕ dz Ω T (Q b α dz u ū Q α dz ϕ dz) <, Q Q where α is defined as (4) α + α + = Some technical lemmas Several technical, analysis lemmas are needed in the paper Our first lemma is a standard iteration lemma which can be found, for example, in [5, Lemma 43] or [3, Lemma 6] Lemma 5 Let φ : [r, R] be a bounded, non-negative function Assume that for all r < s < t R, A φ(t) θφ(s) + (t s) κ + B where A, B, κ > and θ (, ) Then, [ ] A φ(r) C(κ, θ) (R r) κ + B Our next lemma is the classical Hardy s inequality, which can be found, for example, in [6, Theorem 33], [, Lemma 34], and [8]

8 8 T PHAN Lemma 6 Let h : [, ) [, ) be a measurable function such that h(λ)dλ < Then, for every κ, and for every r >, there holds ( ) κ λ r dλ ( κ ) κ h(µ)dµ λ r λ λ r[ λh(λ) ] κ dλ λ The following variant of reverse-hölder s inequality can be found in [, Lemma 35] and it will be useful for the paper Lemma 7 Let h : [, ) [, ) be a non-increasing, measurable function, and let κ [, ), r > Then, there is C > such that ( [ t r h(t) ] ) /κ κ dt λ r h(λ) + C t r h(t) dt, for any λ t t λ 3 Hölder regularity of weak solutions of homogeneous equations We recall some results on Hölder s regularity for weak solutions of homogeneous equations that will be needed in the paper Those results are indeed consequences of the well-known, classical De Giorgi-Nash-Möser theory Our first lemma is about the interior Hölder s regularity estimate, whose proof, for example, can be found in [3, Theorem, p 49 and Theorem p 45], and also in [3, Theorem, Theorem 4] and [45, Theorem ] Lemma 8 Let Λ >, and let A : Q r R n R n be a Carathéodory map and satisfy ()-(3) on Q r with some r > If v is a weak solution of the equation v t div [A (x, t, v)] =, in Q r Then, there exists C > depending only on Λ, n such that v L (Q 5r/6 ) C (Q r v dz) Moreover, there exists β (, ) depending on Λ, n and v L (Q 5r/6 ) such that [ x x v(z) v(z + t t ] β ) C v L (Q 5r/6 ), z = (x, t), z = (x, t ) Q r r/3 To state the boundary regularity, we recall that for some domain Ω R n, and for each r >, z = (x, t ) Ω R, we define Moreover, we also write Ω r (x ) = Ω B r (x ), Ω r = Ω r (), = Ω r (x ) Γ r (t ), K r = K r (, ) T r (z ) = ( Ω B r (x )) Γ r (t ), T r = T r (, ) The following classical boundary Hölder s regularity result can be found in [3, Theorem, p 49 and Theorem p 45], and [45, Theorem 4] Lemma 9 Let Λ > be fixed and let Ω R n be an open bounded domain with boundary Ω C Assume that A : K r R n R n be a Carathéodory map and satisfy ()-(3) on K r R n for some r > Assume also that T r and v is a weak solution of the equation { vt div [A (x, t, v)] =, in K r, A (x, t, v), ν =, on T r, then there exists C > depending only on Λ, n such that v L (K 5r/6 ) C (K r v dz) λ

9 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 9 Moreover, there exists a constant β depending only on Λ, n and v L (K 5r/6 ) such that v C β (K 5r/6 ), and [ x x v(z) v(z + t t ] β ) C v L (K 5r/6 ), z = (x, t), z = (x, t ) K r/3 r 4 Self-improving regularity estimates of Meyers-Gehring s type We need to establish two higher regularity estimates of Meyers-Gehring s type, see [9, 35, 36, 44], for weak solutions of () To begin, let us introduce the following notation with will be used frequently in the paper For each function f defined on U R n+, we write ( ) n (5) G U ( f ) = f n n dz, with n = n + n + 4 Our first lemma is the interior one U Lemma Let Λ > Then, there exists ɛ = ɛ (Λ, n) > such that the following statement holds Suppose that A : Q K R n R n is a Carathéodory map satisfying ()-(3) on Q If u is a weak solution of the equation u t div [A(x, t, λu, u) bu F] = f (x, t), in Q, with some λ Then for every p [, + ɛ ] and γ >, there exists a constant C = C(Λ, p, n) > such that ( /p ( ( /p ( ) u dz) p C u dx) + F dx) p + G p(+γ) (+γ )p dz Q r (z ) Q r (z ) Q r (z ) ( /p +G Q ( f ) f n dx) p, Q r (z ) where z = (x, t ) Q, r (, ), G(x, t) = Ĉ (n, γ )[[u]] BMO(Q,)b with Ĉ (n, γ ) is some definite constant The next lemma is a self-improving regularity estimate on the boundary Lemma For every Λ >, there exists ɛ = ɛ (Λ, n) > such that the following statement holds Suppose that Ω R n with boundary Ω C Suppose that A : K K R n R n is a Carathéodory map satisfying ()-(3) on K K R n and (3) holds on Ω with T Suppose also that u is a weak solution of the equation { ut div [A(x, t, λu, u) bu F] = f (x, t), in K, A(x, t, u, u) bu F, ν =, on T, with some λ Then, for every p [, + ɛ ], and γ >, there exists a constant C = C(Λ, p, γ, n) > such that ( ( ( /p u dz) p C u dz) + K r (z ) ( +G K ( f ) G p(+γ) dz K r (z ) /p f (x, t) n dz) p, K r (z ) ) p(+γ ) ( ) /p + F(x, t) p dz K r (z ) for every z = (z, t ) T, r (, ), G(x, t) = Ĉ (n, γ )[[u]] BMO(K,)b, n = n+ n+4, and with Ĉ (n, γ ) is some definite constant Remark Two remarks on Lemma and Lemma are in ordered (i) Observe that when b L q (Q) and u BMO, it does not follows that ub L q (Q) Therefore, the above self-improving regularity estimates are new and could not directly deduced from the known self-improving regularity estimates

10 T PHAN (ii) If b L (BMO ) and F = f =, a similar self-improving regularity estimate as in Lemma for linear equations is established in [44] From Remark, proofs of Lemma and Lemma are needed We follow the standard approach using Caccioppoli s estimates as in [9, ] Details will be given in the appendix at the end of the paper 3 Approximation estimates 3 Interior approximation estimates In this section, let A : Q R K R n R n satisfy () (4) on Q R K R n for some R > We also recall that p Q R is the parabolic boundary of Q R We study a weak solution u of the class of equations (3) u t div[a(x, t, λu, u) b(x, t)u F] = f (x, t), in Q R, with the parameter λ The following number is used frequently in the paper (3) n = n + n + 4 In the sequel, for each α >, let α > be the number such that α + α =, ie α = α α Moreover, if u is a weak solution of (3), we define G(x, t) = Ĉ (n, α)[[u]] BMO(QR,R)b(x, t), (x, t) Q R, with some definite constant Ĉ (n, α), In our first step, we freeze u in A, and then approximate the solution u of (3) by a solution of the corresponding homogeneous equations with frozen u See also in [5,9,7,38,4,4] for some similar approaches Lemma 3 Let Λ, α > be fixed Assume that A : Q R K R n R n satisfies () (4), and assume that F L (Q R, R n ), f L n (Q R ) and G L α (Q R ) Assume also that u C(Γ R, L (B R )) L (Γ R, W, (B R )) is a weak solution of (3) with some λ Then, for each z = (x, t ) Q R, r (, R), ( ) /n u v dx C(Λ, n) F dz + r f n dz (33) ( + u ū Qr (z ) α dz ) ( α b(x, t) dz) α α, Qr(z) where v C(Γ r (t ), L (B r (x ))) L (Γ r (t ), W, (B r (x ))) is the weak solution of { vt div [A(x, t, λu, v)] =, in Q (34) r (z ), v = u, on p Moreover, it also holds that ( ) v ū Qr (z ) dz (35) ( ( C(n, p) r v u dx) + u ū Qr (z ) dz) Proof Though, the proof is similar and simpler than that of Lemma 35 below We give the proof for the sake of clarity and completeness Observe that for a given weak solution u C(Γ R, L (B R )) L (Γ R, W, (B R )) of (3), by taking A (x, t, ξ) := A(x, t, λu(x, t), ξ), we see that A : Q R R n R n is independent on the variable s K, and it satisfies all assumptions in () (3) Therefore, the existence of weak solution v C(Γ r, L (B r )) L (Γ r, W, (B r )) of (34) can be obtained using the Galerkin s method, [3, p ] It remains to prove the estimates (33), and (35) Through the procedure using

11 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS the Skelov s average (see [5, 3, 38], for examples), we can formally use v u as a test function for the equation (34), and the equation (3), we obtain (36) d dt = B r(x ) B r (x ) v u dx + A(x, t, λu, u) A(x, t, λu, v), v u dx B r (x ) bu + F, u v dx + f (x, t)(v u)dx B r (x ) Also, because div [b(, t)] =, it follows that b(x, t)u(x, t), u v dx B r (x ) = b(x, t)[u(x, t) ū Qr (z )], u v dx B r (x ) ( ) ( u v dx u ū Qr (z ) α dx B r (x ) B r (x ) ) /α ( ) /α b(x, t) α B r (x ) Then, it follows from an integration in time, Remark, (36), and the Young s inequality that sup Γ r (t ) B r(x ) C(Λ) sup v u dx + Γ r (t ) B r(x ) v u dx + u v dz [ C(Λ) bu + F, u v dz + Q r (x ) { u v dz + C(Λ) ( ) /α ( + u ū Qr (z ) α dxdt A(x, t, λu, u) A(x, t, λu, v), v u dz F dz + ] f v u dz f v u dz /α b(x, t) α dxdt) Hence, (37) sup r Γ r (t ) { C(Λ) + B r(x ) v u dx + F dz + ( u ū Qr (z ) α dxdt u v dz f v u dz ) /α ( /α b(x, t) α dxdt) Now, let us denote p = n > where n is the number defined in (3) Also, let p be the number such that p + p =, ie p = (n+) n It follows from Hölder s inequality, and the parabolic Sobolev imbedding

12 T PHAN (see [3, eqn (3), p 74] or [3, Proposition 3, p 7]), and Young s inequality ( ) /p ( ) /p f v u dz v u p dz f p dz (38) ( C(n)r 4 /p v u dz) sup r Γ r (t ) B r (x ) v u dx np ( f p dz ) /p v u) dz + ( /p 4 sup r v u dx + C(n)r f p dz) Γ r (t ) B r (x ) The estimate (38), and (37) imply that sup r v u dx + u v dz Γ r (t ) B r(x ) ( ) /p C (Λ, n) F dz + r f p dz ( + ) u ū Qr (z ) α dxdt ( α b(x, t) dxdt) α α Qr(z) This proves the estimate (33) Also, by the Poincaré s inequality, we see that ( ( ( v ū Qr (z ) dz) v u dz) + u ū Qr (z ) dz) ( ( C(n, p)r v u dz) + u ū Qr (z ) dz) Q r (x ) This proves (35) and completes the proof of the lemma The next step is to approximate the solution u in Q κr (z ) by the solution w of { wt div [A(x, t, λū (39) Qκr (z ), w)] =, in Q κr (z ), w = v, on p Q κr (z ), where in (39) v is the weak solution of (34), and κ (, /3) is some sufficiently small constant which will be determined Lemma 3 Let Λ, M >, α >, and α (, ] be fixed, and let ɛ (, ) There exist positive, sufficiently small numbers κ = κ(λ, M, n, α, ɛ) and δ = δ (ɛ, Λ, M, n, α ) (, ɛ) such that the following holds Assume that A : Q R K R n R n satisfies () (4), and assume that F L (Q R, R n ), G L α (Q R, R n ), f L n (Q R ) and ( /n ( ) F dz + r f n dz) + G(x, t) α α dz δ, for some z = (x, t ) Q R and some r (, R) Then, for every λ >, if u C(Γ R, L (B R )) L p (Γ R, W, (B R )) is a weak solution of (3) satisfying u dz, and [[λu]] BMO(QR,R) M, it holds that Q κr (z ) where w is the weak solution of (39) Q κr (z ) v w dz ɛ,

13 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 3 Proof The proof is similar that of Lemma 36 in the next subsection using Lemma 8 and (35) instead of Lemma 9 and (38) respectively We therefore skip the proof The next lemma is a general result which particularly compares the solution w of (39) with a solution of the corresponding constant coefficient equation Lemma 33 Let Λ > be fixed, then there is some γ = γ(λ, n) > such that the following statement holds For some z = (x, t ) Q R, assume that A : Q R R n R n such that () (3) hold for some R > Assume also for some ρ (, R/), w is a weak solution of w t div [A (x, t, w)] =, in Q ρ (z ) Then, there is some function h L ((Γ 7ρ/4 (t ), L (B 7ρ/4 (x )) L (Γ 7ρ/4 (t ), W, (B 7ρ/4 (x ))) such that w h dz C(Λ, n)[a ] /γ Q 7ρ/4 (z ) BMO(Q R,R) w dz Q ρ (z ) Moreover, Q 7ρ/4 (z ) h L (Q 3ρ/ (z )) C(Λ, n) Q ρ (z ) Q ρ (z ) w dz Q ρ (z ) Proof The proof is simple, and we give it here for the sake of completeness Observe that from Lemma, there is p = p (Λ, n) > such that (3) w p dz /p C(Λ, p, n) w dz Q 7ρ/4 Q ρ Let us denote a(t, ξ) = B 7ρ/4 (x ) A (x, t, ξ)dx, Θ A,B 7ρ/4 (x ) = A (x, t, ξ) a(t, ξ), ξ R n \ {} ξ Then, let h be the weak solution of { ht div [a(t, v)] =, in Q (3) 7ρ/4 (z ), h = w, on p Q 7ρ/4 (z ) Observe that the existence of h can be obtained by a standard method using Galerkin s approximation Also, from the Skelov s average as in [5, 3], we can formally use w h as a test function for both the equations of w and of h to obtain d w h dx + a(t, w) a(t, h), w h dx dt B 7ρ/4 (x ) B 7ρ/4 (x ) = a(t, w) A (x, t, h), w h dx B 7ρ/4 (x ) This and () imply that sup t Γ 7ρ/4 (t ) C(Λ) C(Λ) B 7ρ/4 (x ) Q 7ρ/4 (z ) Q 7ρ/4 (z ) w h dx + w h dz Q 7ρ/4 (z ) a(t, w) A (x, t, w) w h dz Θ A,B 7ρ/4 (x )(x, t) w w h dz Let us now denote γ > be a number satisfying γ + p =

14 4 T PHAN Then, by using Hölder s inequality and (3), we see that w h dz C(Λ) Θ A,Ω 7ρ/4 (x )(x, t) γ dz Q 7ρ/4 (z ) Q 7ρ/4 (z ) Q 7ρ/4 (z ) C(Λ, n)[a ] /γ BMO(Q R,R) w h dz Q ρ (z ) w dz /γ Q 7ρ/4 (z ) Q 7ρ/4 (z ) w p dz /p w h dz Hence, w h dz C(Λ, n)[a ] /γ BMO(Q R,R) w dz Q 7ρ/4 (z ) Q ρ (z ) and this proves the first assertion of the lemma To prove the last estimate of Lemma 33, we can use standard regularity theory for equation (3) to obtain h L (Q 3ρ/ (z )) C(Λ, n) h dz Q 7ρ/4 (z ) This, together with the fact that [A ] BMO(QR,R) C(Λ, n), triangle inequality, and (333) imply (3) The proof of the lemma is then complete Our next result is the main result of the section Proposition 34 Let Λ >, α > and α (, ] be fixed Then, for every ɛ (, ), there exist sufficiently small numbers κ = κ(λ, M, n, α, ɛ) (, /3) and δ = δ (ɛ, Λ, M, n, α ) (, ɛ) such that the following holds Assume that A : Q R K R n R n satisfies () (4) and () hold with δ replaced by δ Assume also that ( /n ( ) F dz + r f n dz) + G(x, t) α α dz δ, Q r (z ) Q r (z ) Q r (z ) for some z = (x, t ) Q R and some r (, R/) Then, for every λ, if u C(Γ R, L (B R )) L (Γ R, W, (B R )) is a weak solution of (3) satisfying u dz, and [[λu]] BMO(QR,R) M, Q 4 κr (z ) then there is h C(Γ 7 κr/ (t ), L (B 7 κr/ (x ))) L (Γ 7 κr/ (t ), W, (B 7 κr/ (x ))) such that the following estimate holds (3) u h dz ɛ, h Q 7 κr/ (z ) L (Q 3 κr (z )) C(Λ, n) Q 7 κr/ (z ) Proof The proposition follows directly by applying Lemma 3 with r replaced by r, and Lemma 33 with A (x, t, ξ) = A(x, t, λū Q4 κr (z ), ξ) and ρ = κr, and the triangle inequality 3 Boundary approximation estimates To be convenient for the readers and self-contained, we recall some frequently used notation For each R >, we write B R = B R () the ball in R n centered at the origin with radius R Moreover, for an open set Ω R n with boundary Ω, we write Ω R = B R Ω, K R = Ω R Γ R, and T R = ( Ω B R ) Γ R We also denote p K R the parabolic boundary of K R, moreover, for each z = (x, t ), it is denoted that Ω R (x ) = x + Ω R, K R (z ) = z + K R, T R (z ) = z + T R

15 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 5 We can assume that T R, and we investigate weak solutions u of the equation { ut div [A(x, t, λu, u) b(x, t)u F(x, t)] = f (x, t), in K (33) R, A(x, t, λu, u) b(x, t)u F(x, t), ν =, on T R with the parameter λ By weak solutions of (33), we mean any u C(Γ R, L (Ω R )) L (Γ R, W, (Ω R )) such that G L α (K R ) for some α >, λu(x, t) K for ae (x, t) K R, and u(x, t) t ϕ(x, t)dz + A(x, t, λu, u) bu F, ϕ dz = f (x, t)ϕ(x, t)dz, K R K R K R for all ϕ C (Q R) Here, (34) G(x, t) = Ĉ (n, α)[[u]] BMO(KR,R)b(x, t), z = (x, t) K R, for some definite constant Ĉ (n, α) As before, for each α >, let α be the number satisfying (35) α + α =, ie α = α α Recall that the number n is defined in (3) Our first step is to approximate u by the solution v of the homogeneous equation with frozen u in A Lemma 35 Let Λ, α > be fixed Assume that A : K R K R n R n satisfies () (4) Assume that G L α (K R ), and u is a weak solution of (33) for some λ, then for each z = (x, t ) K R and each r (, R), it holds that ( ) u v dz C (Λ, n) F dz + G(x, t) α α ( /n dz + r f n dz) (36), where v C(Γ r (t ), L (Ω r (x ))) L (Γ r (t ), W, (Ω r (x ))) is the weak solution of v t div [A(x, t, λu, v)] =, in, (37) v = u, on p \ T r (z ), A(x, t, λu, v), ν =, on T r (z ), if T r (z ) Moreover, (38) ( ( v ū Kr (z ) dz) C(n) r ( u v dz) + u ū Kr (z ) dz) Proof If T r (z ) =, this lemma follows directly from Lemma 3 Therefore, we only consider the case that T r (z ) The proof is similar to that of Lemma 3 with some modification dealing with the boundary Observe that since Ω C, Ω r (x ) is a Lipschitz domain Therefore, W, (Ω r (x )) is well-defined with all imbdedding and compact imbedding properties Therefore, the existence of the solution v of (38) can be obtained by the Galerkin s method, see [3, p ] It then remains to prove the estimates (36) and (38) By proceeding with the Steklov s average (see [5, 3, 38]), we can formally use v u as a test function for the equations (37), and (33) to infer that d v u dx + A(x, t, λu, u) A(x, t, λu, v), u v dx dt Ω r (x ) Ω r (x ) = b(x, t)u + F, u v dx + f (v u)dx Ω r (x ) Ω r (x ) Due to the fact that div b = on Ω R and b, ν = on B R Ω in the sense that b(x, t) ϕ(x)dx =, for all ϕ C (B r(x )), for ae t Γ R, Ω r (x )

16 6 T PHAN we see that Therefore, d dt = Ω r (x ) Ω r (x ) Ω r (x ) b(x, t)u, u v dx = Ω r (x ) b(x, t)[u ū Kr (z )], u v dx v u dx + A(x, t, λu, u) A(x, t, λu, v), u v dx Ω r (x ) b(x, t)[u ū Kr (z )] + F, u v dx + f (v u)dx From this, and the condition () (4), we infer that sup v u dx + u v dz t Γ r (t ) Ω r (x ) [ sup v u dx + C(Λ) Then, t Γ r (t ) Ω(x ) [ ( C(Λ) F + b u ūkr (z ) ) u v + u v dz [ + C(Λ) sup r t Γ r (t ) [ C(Λ, n) Ω r (x ) A(x, t, λu, u) A(x, t, λu, v), u v dz ] f v u dz ( F + b u ū Kr (z ) ) ] dz + f v u dz Ω r (x ) v u dx + u v dz ( F + b u ū Kr (z ) ) dz + ] f v u dz Now now control the last two terms in the right hand side of the previous estimate Observe that from (35), the John-Nirenberg s theorem, and the Hölder s inequality it follows that b u ū Kr (z ) ) dz ( ) /α ( C(n) b α dz ( Ĉ (n, α)[[u]] BMO(K R,R) /α b dz) α = ) /α u ū Kr (z ) α dz ( /α G(x, t) dz) α On the other hand, as in (38), we denote p = n and p such that /p + /p =, ie p = n n+ From Hölder s inequality, the parabolic Sobolev imbedding (see [3, eqn (3), p 74] or [3, Proposition 3, p 7]), and Young s inequality, it follows that ( ) /p ( ) /p f v u dz v u p dz f p dz (39) v u dz + ( /p 4 sup r v u dx + C(n)r f p dz) Γ r (t ) B r (x ) ]

17 Therefore, (3) LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 7 sup r t Γ r (t ) C(Λ, n) Ω r (x ) v u dx + u v dz ( F + G(x, t) α dz ) /α + r ( /p f (x, t) p dz) and (36) follows Lastly, we prove (38) Observe that the triangle inequality gives [ ] v ū Kr (z ) dz C v u dz + u ū Kr (z ) dx K r (x ) Then, using the Poincaré s inequality for the first term in the right hand side of the above inequality, we see that ( ( ( v ū Kr (z ) dz) C(n) r u v dz) + u ū Kr (z ) dz) This proves (38) and also completes the proof Lemma 36 Let Λ, M >, α > and α (, ) be fixed Then, for every ɛ (, ), there exist sufficiently small numbers κ = κ(λ, M, n, α, ɛ) (, /3) and δ = δ (ɛ, Λ, M, n, α ) (, ɛ) such that the following holds Assume that A : K R K R n R n such that () (4) hold with some R > and some open set K R, and assume that ( ) (3) F dz + G α α ( ) /n dz + r f n dz δ, for some z = (x, t ) K R and some r (, R) Then, for every λ, if u is a weak solution of (33) satisfying u dx, and [[λu]] BMO(KR,R) M, K κr (z ) then there is a weak solution w of w t div [A(x, t, λū Kκr (z ), w)] =, in, (3) w = v, on p \ T κr (z ), A(x, t, λū Kκr (z ), w), ν =, on T κr (z ), if T κr (z ), such that the following estimate holds ( ( ) (33) u w dz) ɛ, and w dz + n+ where in (3) the function v is defined as in Lemma 35 Proof For a given sufficiently small ɛ >, let ɛ (, ɛ/) and κ (, /3) both sufficiently small depending on ɛ, Λ, M, n, α which will be determined Now, by Lemma 35 with ɛ, we can find δ = δ (ɛ, Λ, κ) > sufficiently small such that if (3) holds, then ( ) (34) u v dz (ɛ ) κ n+, and λ v ū Kr (z ) dz C(n, p)[rɛ κ n+ λ + M], where v is the solution of (37) Observe that the first inequality in (34) and the fact that ɛ, κ (, ) imply ( ( ( ) v dz) u v dz) + u dz K κr (z ) K κr (z ) K κr (z ) (35) ( (κ) n+ u v dz) +

18 8 T PHAN Note that when λ =, w = v The lemma is then trivial with every κ (, /3) Therefore, we only need to consider the case λ > From the standard Caccioppli s type estimate for the solution v of (37), and (34), we also see that ( ( (36) v dz) K κr (z ) C(Λ, n) ( κ)κ n+ r v ū Kr (z ) dz C(Λ, n) [ ɛ + M(λκ n+ r) ], where in the last estimate we have used the fact that κ (, /3) to control the factor κ Now, let w be the weak solution of (3), whose existence can be obtained by a standard method using Galerkin s method, see [3, p ] It remains to prove the estimate (33) By using the Skelov s average as in [5, 3], we can formally take v w as a test function for the equation (3) and the equation (37) to obtain d v w dx + A(x, t, λu, v), w v dx dt Ω κr (x ) Ω κr (x ) (37) = A(x, t, λū Kκr (z ), w), w v dx Ω κr (x ) From this, it follows that sup v w dx + v w dz Γ κr (t ) Ω κr (x ) C(Λ) sup v w dx + Γ κr (t ) Ω κr (x ) ) ] A(x, t, λū Kκr (z ), v) A(x, t, λū Kκr (z ), w), v w dz [ C(Λ) A(x, t, λū Kκr (z ), v), v w dz ] + A(x, t, λu, v), v w dz C(Λ) v v w dz C(Λ) v dz + v w dz This last estimate together with (36) imply that ( ( ) v w dz) C(Λ, n) v dz ( ) C(Λ, n) v dz C (Λ, n) [ ɛ + M(rκ ] n+ λ) K κr (z ) Hence, if MC (Λ, n)(λκ n+ r) ɛ 4, we can choose ɛ sufficiently small such that (38) 4C (Λ, n)ɛ < ɛ From these choices, it follows that ( v w dz) ɛ/

19 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 9 This estimate, the triangle inequality, and the first estimate in (34) gives ( ( ( u w dz) u v dz) + v w dz ( κ n+ u v dz) + ɛ/ ɛ + ɛ/ ɛ which is the first estimate in (33) It therefore remains to consider the case (39) λrκ n+ ɛ C (Λ, n)m In this case, we note that from our choice that ɛ ɛ, we particularly have λκ n+ ɛ r C(Λ, M, n) Then, it follows from (34) that ( λ v ū Kr (z ) dz) C(Λ, M, n) K r (x ) From this and the equation (37), we can apply the Hölder s regularity theory in Lemma 8 and Lemma 9 for the function ṽ(x, t) := λ[v(x x, t t ) ū Kr (z )], (x, t) K r, to find that there is β (, ) depending only on Λ, n such that ṽ C β ( K r/3 ) Then, by scaling back, we obtain the following estimates λ v ūkr (z ) L C(Λ, M, n), (K 5r/6 (z )) and (33) [ x x λ v(z) v(z + t t ] β ) C(Λ, M, n) κ β, z = (x, t), z = (x, t ) K κr(z ) r From now on, for simplicity, we write û = u ū Kκr (z ) We can use (37) again to obtain sup v w dx + v w dz Γ κr (t ) Ω κr (x ) C(Λ) sup v w dx Hence, Γ κr (t ) Ω κr (x ) ] + A(x, t, λū Kκr (z ), v) A(x, t, λū Kκr (z ), w), v w dz C(Λ) A(x, t, λū Kκr (z ), v) A(x, t, λu, v), v w dz C(Λ) [λû] α v v w dz v w dz + C(Λ) λû α v dz v w dz C(Λ) λû α v dz )

20 T PHAN For p > and sufficiently close to depending only on Λ, n, we write q = α p p > Then, using the Hölder s inequality, and the self-improving regularity estimate, Lemma, we obtain We further write λû q dz = Therefore, This, and (35) imply that (33) ( ) p ( ) v w dz C(Λ) λû q p v p p dz ( ( C(Λ, n) λû q dz ( û λû q dz λû dz ( C(n, α )[[λu]] q BMO(K R,R) ) ( ) p p v dz K κr (z ) λû q dz ( λû dz) C(n, M, α ) ) ) ( ) p ( ) v w dz C(Λ, M, n, α ) λû p dz v dz K κr (z ) ( ) p v w dz C(Λ, M, n, α ) λû p dz On the other hand, we also write [ λû dz C λ(u v) dz + λ(v v Kκr (z )) dz + λ(ū Kκr (z ) v Kκr (z )) dz C(n) [κ (n+) λ(u v) dz + ] λ(v v Kκr (z )) dz λû dz) Then, by using the Poincaré s inequality for the first term on the right hand side of the last estimate, we obtain ( ( λû dz) n+ C(Λ, n) λrκ u v dz) + λ sup v(z) v(z ), x,y This, (34), and (33) imply that ( λû dz) C(Λ, n) [ (λr)ɛ + κ ] β From this last estimate, the estimate (33) can be written as v w dz C(Λ, M, n, α ) ( ) λrɛ + κ β p p This, (39), we can further imply that ( v w dz ) C (Λ, M, α, n) [ ɛ ɛκ n+ + κ β ] p p

21 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS Now, we choose κ sufficiently small depending only on Λ, M, n, α and ɛ such that κ β [ ɛ ] p p C (Λ, M, α, n) Then, we choose ɛ > sufficiently small depending only on Λ, n, α and ɛ such that ɛ n+ ɛκ [ ɛ ] p p C (Λ, M, α, n) From these choices, we obtain ( v w dz) ɛ/ Then, we use the first estimate in (34), the triangle inequality again to obtain the first estimate in (33) It now remains to prove the second estimate in (33) By the triangle inequality, the assumption in this lemma and since ɛ (, ), we see that ( ( ( w dz) ɛ + as desired The proof is therefore complete u dz) + u dz ) ( n+ w dz + n+ K κr (z ) Our next result is a standard approximation which particularly gives a comparison of the solution w of (3) with a constant coefficient solution Lemma 37 Let Λ > be fixed Then, for every ɛ (, ), there exists a constant δ = δ (Λ, n, ɛ) > and sufficiently small such that the following statement holds Assume that Ω is be an open bounded domain with boundary Ω C Assume also that A : K R R n R n satisfying () (3) and [A ] BMO(KR,R) δ for some R > Suppose that w is a weak solution of { wt div [A (33) (x, t, w)] =, in K 4ρ (z ), A (x, t, w), ν =, on T 4ρ (z ), if T 4ρ (z ), and it satisfies w dz, K 4ρ with some < ρ < R/4, and some z = (x, t ) K R Then, there is some function h such that (333) w h dz ɛ, and h L (K ρ (z )) C(Λ, n) Proof The proof can be done exactly the same as that of Lemma 33 Since Ω is C, the Lipschitz regularity estimates for weak solutions of the corresponding homogeneous equation with frozen coefficient holds true if T ρ (z ) Alternatively, since Ω is C, it is sufficiently flat in the sense of Reifenberg s Therefore, this lemma follows from [8, Lemma 6 and Corollary ], see also [6] One can flatten the boundary as in [7] and prove a similar approximation in the upper-half cube Q + r as Lemma 33 Finally, we state and prove the main result of the section Proposition 38 Let Λ, M >, α > and α (, ) be fixed Then, for every ɛ (, ), there exist sufficiently small numbers κ = κ(λ, M, n, α, ɛ) (, /3) and δ = δ(ɛ, Λ, M, n, α ) (, ɛ) such that the following holds Assume Ω C, and A : K R K R n R n such that () (4) and (4) hold, and assume that ( ) F dz + G α α ( /n dz + (8r) f n dz) δ, K 8r (z ) K 8r (z ) K 8r (z ) )

22 T PHAN for some z = (x, t ) K R, and r (, R/8) Then, for every λ, if u is a weak solution of (33) satisfying u dz, and [[λu]] BMO(KR,R) M, K 6κr (z ) then there is h L (Γ 4κr (t ), L (Ω 4κr (x ))) L (Γ 4κr (t ), W, (Ω 4κr (x ))) such that the following estimate holds (334) u h dz ɛ, h L (K κr (z )) C(Λ, n) K 4κr (z ) Proof Let δ = min { δ ( [ ]n+ ɛ, Λ, M, n, α), δ (Λ, n, ɛ/[( + (n+)/ )]) }, ] where δ is defined in Lemma 36, and δ is defined in Lemma 37 Moreover, let κ = κ(λ, M, n, α, [ n+ ɛ) be the number defined in Lemma 36 Then, by applying Lemma 36 with r replaced by 8r, we can find w L (Γ 8κr (t ), L (Ω 8κr (x ))) L (Γ 8κr (t ), W, (Ω 8κr (x ))) satisfying ( ) (335) u w dz ] ( ) n+ K 8κr (z ) [ ɛ and w dz + n+ K 8κr (z ) Then, we can apply Lemma 37 with A (x, t, ξ) = A(x, t, λū K8κr (z ), ξ), ρ = κr, and with some suitable scaling, we can find a function h L (Γ 4κr (t ), L (Ω 4κr (x ))) L (Γ 4κr (t ), W, (Ω 4κr (x ))) such that the following estimate holds ( (336) w h dz) ɛ/, h L (K κr (z )) C(Λ, n) K 4κr (z ) It then follows from (335), (336), and the triangle inequality that ( ( u h dz) u w dz) + K 4κr (z ) K 4κr (z ) The proof is therefore complete [ ] n+ ( ( w h dz K 4κr (z ) u w dz) + ɛ/ ɛ K 8κr (z ) 4 Level set estimates This section gives the key level set estimates needed in the proofs of the main theorems, ie Theorem, and Theorem We can assume R = as Theorem, and Theorem can be retrieved to general R > by using the dilation (3), Remark, and the dilation property of Lorentz quasi-norms, Remark Let ɛ > be a sufficiently small number to be determined depending only on the given numbers n, Λ, p, q, and α Let δ = δ(ɛ, Λ, M, n, α ), κ = κ(λ, M, n, α, ɛ) be the numbers defined in Proposition 38 Note that since ɛ depends on n, Λ, p, q and α, the numbers κ and δ also only depend on these numbers Assume that all assumptions in Theorem are valid with this δ and R = For each λ, and if u is a weak solution of (5), recall that G(x, t) [[u]] BMO(K,)b(x, t), (x, t) K We fix η > such that η < min{ + ɛ, p}, where ɛ = ɛ (Λ, n) validates both Lemma and Lemma Let us also denote (4) F(x, t) = F(x, t) + G(x, t) + G( f ) f (x, t) n, (x, t) K, )

23 LORENTZ ESTIMATES FOR WEAK SOLUTIONS, SINGULAR DIVERGENCE-FREE DRIFTS 3 where n is defined in (3), and ( (4) G( f ) = f (x, t) n dz K As we will see in (4) and (43) below, the function G plays an essential role in our proof Observe also that since p (43) G( f ) f n L p,q (K ) = f n L n (K ) f n L n p,n q (K ) C(n) f L n p,n q (K ) From now on, let τ > be the number defined by ( ) (44) τ = u dz + ( ) /η F η dz < K δ K For fixed numbers µ, and τ >, we denote the upper-level set of u in K µ by ) n n (45) E µ (τ) = { Lebesgue point (x, t) K µ of u : u(x, t) > τ } The following Proposition estimating the upper-level sets of u is the main result of this section Proposition 4 There exist N = N (Λ, n) > and B = B (n) such that [ E s (N τ) ɛ E s (τ/4) + (δτ) η δτ/4 s η {(x, t) K : F(x, t) > s} ds s for all s < s, for every τ > ˆB τ, where ˆB := B [(s s )κ] n+ The rest of the section is to prove this proposition We follow the approach developed in [] and used in [, 5, 6] However, some nontrivial modifications are also required to treat the terms f, b and to obtain the sharp homogeneous estimates, see Remark 4 For each z K and each r >, we define ( ( (46) CZ r ( z) = u dz K r ( z) ) + δ /η F dz) η K r ( z) Several lemmas are needed to prove Proposition 44 Our first lemma is a stopping-time argument lemma Lemma 4 There exists a constant B = B (n) such that for each s < s, τ > ˆB τ, and for z E s (τ), there is r z < (s s )κ 4 such that CZ r z ( z) = τ, and CZ r ( z) < τ, r (r z, ) Proof The argument is quite standard, see [,, 5, 6] Observe that because r <, and η >, we have ( ) (n+)/ ( ) ( ) (n+)/η ( ) /η CZ r ( z) C(n) u dz + F η dz r K r δ C(n)τ K r (n+)/ Therefore, if r > (s s )κ 4, then for B = C(n)[4] (n+)/, we see that ( ) (n+)/ C(n)τ 4 C(n) τ r (n+)/ = B [(s s )κ] n+ τ < τ (s s )κ Then, (s s )κ CZ r ( z) < τ, when r, and τ > ˆB τ 4 On the other hand, when z E s (τ), by the Lebesgue s theorem, we see that if r is sufficiently small, then CZ r ( z) > τ Due to the fact the CZ r ( z) is absolutely continuous, we can find r z, which is the largest number in (, (s s )κ 4 ), such that CZ r z ( z) = τ From this, the conclusion of the lemma follows ],