Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

(1)

www.elsevier.com/locate/anihpc

Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

Susan Friedlander

^a,^∗

, Vlad Vicol

^b

aDepartment of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089, United States bDepartment of Mathematics, University of Chicago, 5734 University Ave., Chicago, IL 60637, United States

Received 10 July 2010; received in revised form 3 January 2011; accepted 11 January 2011 Available online 31 January 2011

Abstract

We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with L² initial data and divergence free drift velocity that lies inL^∞_t BMO⁻_x¹. We apply this result to prove global regularity for a family of active scalar equations which includes the advection–diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth’s fluid core.

Résumé

Nous utilisons des techniques de De Giorgi pour démontrer la continuité Hölder de solutions faibles pour une classe d’équations de dérive-diffusion, avec données initialesL²et champ de vitesse incompressible appartenant àL^∞_t BMO⁻_x¹. Nous appliquons ce résultat pour démontrer la régularité globale pour une famille d’équations du scalaire actif qui comprend l’équation d’advection–

diffusion qui a été proposée par Moffatt dans le contexte de la turbulence magnétostrophique dans le noyau fluide de la Terre.

MSC:76D03; 35Q35; 76W05

Keywords:Global regularity; Weak solutions; De Giorgi; Parabolic equations; Magneto-geostrophic equations

1. Introduction

Active scalar evolution equations have been a topic of considerable study in recent years, in part because they arise in many physical models. In particular, such equations are prevalent in fluid dynamics. In this paper we first examine a class of drift-diffusion equations for an unknown scalar fieldθ (t, x), of the form

∂_tθ+(v· ∇)θ=θ, (1.1)

wherev(t, x)is a given divergence free vector field that lies in the function space L²_tL²_x∩L^∞_t BMO⁻_x¹,t >0, and x∈R^d. In Theorem 2.1 we prove that weak solutions to (1.1) are Hölder continuous. Note that this result is new for

* Corresponding author.

E-mail addresses:susanfri@usc.edu (S. Friedlander), vicol@math.uchicago.edu (V. Vicol).

doi:10.1016/j.anihpc.2011.01.002

(2)

such linear parabolic equations with very singular coefficients [1,18,19,25,27,34,35]. We then use this result to prove in Theorem 3.1 that Leray–Hopf weak solutions of the active scalar equation

∂_tθ+(u· ∇)θ=θ, (1.2)

divu=0, (1.3)

uj=∂iTijθ (1.4)

are classical solutions. In (1.4), the velocity vectoruis obtained fromθvia{T_ij}, ad×dmatrix of Calderón–Zygmund singular integral operators (that is, they are boundedL²→L²andL^∞→BMO) such that∂_i∂_jT_ij≡0. Note that in (1.4) we have used the summation convention on repeated indices, andi, j∈ {1, . . . , d}.

Our motivation for addressing the system (1.2)–(1.4) comes from a model proposed by Moffatt [22] for magnetostrophic turbulence in the Earth’s fluid core. This model is derived from the full magnetohydrodynamic equations (MHD) in the context of a rapidly rotating, density stratified, electrically conducting fluid. After a series of approxima- tions relevant to the geodynamo model, a linear relationship is established between the velocity and magnetic vector fields, and the scalar “buoyancy”θ. The sole remaining nonlinearity in the system occurs in the evolution equation forθ, which has the form

∂tθ+(u· ∇)θ=S+κθ, (1.5)

whereSis a source term, andκis the coefficient of thermal diffusivity. Here the three-dimensional velocityuis such that divu=0, and it is obtained from the buoyancy via

u=M[θ], (1.6)

whereMis a nonlocal differential operator of order 1. We describe the precise form of the operatorMin Section 4.

An important feature of this operator is the spatial inhomogeneity that occurs due to the underlying mean magnetic field. We call (1.5)–(1.6) themagnetogeostrophic equation(MG). We show that the MG system satisfies the conditions under which we prove Theorem 3.1, and hence obtain (cf. Theorem 4.1) global well-posedness for (1.5)–(1.6).

An active scalar equation that has received much attention in the mathematical literature following its presentation by Constantin, Majda, and Tabak [9], as a two-dimensional toy model for the three-dimensional fluid equations, is the so called surface quasi-geostrophic equation (SQG) (see, for example, [2,6,7,10,11,16,29,33] and references therein).

The dissipative form of this equation for which there is a physical derivation is

∂tθ+(u· ∇)θ= −(−)^1/2θ, (1.7)

where

u= ∇^⊥(−)⁻^1/2θ≡(R₂θ,−R₁θ ) (1.8)

andR_i represents theith Riesz transform. It was recently proved by Caffarelli and Vasseur [2] that solutions of (1.7)–

(1.8) withL²initial data are smooth (see also the review article [3]). Well-posedness for (1.7)–(1.8) in the case of smooth periodic initial data was also obtained by Kiselev, Nazarov, and Volberg [16]. See also Constantin and Wu [10, 11] for the super-critically dissipative SQG.

We note that the magnetogeostrophic equation MG and the critically dissipative SQG equation (1.7)–(1.8) are both derived from the Navier–Stokes equations in the context of a rapidly rotating fluid in a thin shell. For both systems the Coriolis force is dominant in the momentum equation. In the case of the SQG equation the relation (1.8) is derived via a projection of the three-dimensional problem onto the two-dimensional horizontal bounding surface. In the case of the MG equation the coupling with the magnetic induction equation closes the three-dimensional linear system that produces the operators{T_ij}, withu_j=∂_iT_ijθ.

Systems (1.2)–(1.4) and (1.7)–(1.8) have strong similarities. In particular, they have the same relative order of the spatial derivatives between the advection term and the diffusive term. Moreover, ifθ (t, x)is a solution of (1.2)–

(1.4), thenθ_λ(t, x)=θ (λ²t, λx)is also a solution, and henceL^∞(R^d)is the critical Lebesgue space with respect to the natural scaling of the equation. We note thatL^∞is also the critical Lebesgue space for the critically dissipative surface quasi-geostrophic equation (1.7)–(1.8), and for the modified surface quasi-geostrophic equation (cf. Constantin, Iyer, and Wu [8]). The advantage of system (1.2)–(1.4) over the critical SQG equation is that the diffusive term is given

(3)

via a local operator. The tradeoff is that the drift velocity in (1.2)–(1.4) is more singular, i.e., the derivative of aBMO function (see Koch and Tataru [17] for the Navier–Stokes equations inBMO⁻¹).

Our proof of Theorem 2.1 and Theorem 3.1 is along the lines of the proof of Caffarelli and Vasseur [2, Theorem 3]

for the critical SQG equation. The primary technique employed in [2,11,32], and in the present paper, is the De Giorgi iteration [12]. This consists of first showing that a weak solution is bounded by proving that the function max{θ−h,0}

has zero energy ifhis chosen large enough. Then a diminishing oscillation result implies smoothness of the solution in a subcritical space, namely C^α, for someα∈(0,1). The proof of Hölder continuity for solutions of (1.1) with v∈L^∞_t BMO⁻_x¹does not follow directly either from [2], wherev∈L^∞_t BMOx, or from [8], wherev∈L^∞_t C_x¹⁻^α and α∈(0,1). The crucial step in the proof of Theorem 2.1 is the local energy and uniform estimates. The main obstruction to applying the classical parabolic De Giorgi estimates via anL^p-based Caccioppoli inequality (1< p <∞), is that v(t,·)∈BMO⁻¹. In Section 2 we give details as to how we overcome this difficulty.

Eq. (1.1) is in the class of parabolic equations in divergence form that have been studied extensively, including in the classical papers of Nash [24], Moser [23], Aronson and Serrin [1]. Osada [25] allowed for singular coefficients and proved Hölder continuity of solutions to (1.1) whenv∈L^∞_t W_x⁻^1,^∞is divergence free. Hence Theorem 2.1 may be also viewed as an improvement of the results of Osada, since iff∈BMO(R^d)∩L²(R^d), then it does not follow that f ∈L^∞(R^d)(cf. [30]). In the same spirit, Zhang [34,35] and Semenov [27] give strong regularity results for parabolic equations of the type (1.1), where the singular divergence free velocity satisfies a certain form boundedness condition.

We note that this form boundedness condition does not cover the casev∈L^∞_t BMO⁻_x¹, and hence Theorem 2.1 does not follow from the results in [27,34,35], and vice-versa. The overall conclusion of the body of work on parabolic equations with a singular drift velocity is that the divergence free structure ofvproduces a dramatic gain in regularity of the solution, compared to the classical theory (cf. [18]).

Organization of the paper. In Section 2 we prove Hölder regularity for the linear drift-diffusion equation (1.1), with vbeing a given divergence free vector field in the function spaceL²_t,x∩L^∞_t BMO⁻_x¹. In Section 3 we apply this result to prove that a Leray–Hopf weak solutionθ of the nonlinear active scalar system (1.2)–(1.4) is Hölder smooth for positive time. Since Hölder regularity is subcritical for the natural scaling of (1.2)–(1.4) we can bootstrap to prove higher regularity and hence conclude that the solution is a classical solution. In Section 4 we describe an active scalar equation that arises as a model for magneto-geostrophic dynamics in the Earth’s fluid core. We show that this three- dimensionalMG equationis an example of the general system (1.2)–(1.4). In Appendix A we prove the existence of weak solutions to (1.2)–(1.4) evolving fromL²(R^d)initial data.

2. Regularity for a parabolic equation with singular drift Consider the evolution of an unknown scalarθ (t, x)given by

∂_tθ+(v· ∇)θ=θ (2.1)

where the velocity vector v(t, x)=(v₁(t, x), . . . , v_d(t, x))∈L²((0,∞)×R^d)is given, and(t, x)∈ [0,∞)×R^d. Additionally letvsatisfy

∂_jv_j(t, x)=0 (2.2)

in the sense of distributions. We expressv_jas

v_j(t, x)=∂_iV_ij(t, x) (2.3)

in [0,∞)×R^d, where we have used the summation convention on repeated indices, and we denoted V_ij =

−(−)⁻¹∂_iv_j. The matrix{V_ij}^d_i,j₌₁is given, and satisfies V_ij∈L^∞

(0,∞);L² R^d

∩L²

(0,∞); ˙H¹ R^d

(2.4) for alli, j∈ {1, . . . , d}.

Theorem 2.1(The linear problem). Given θ₀∈L²(R^d)and {V_ij} satisfying(2.4), let θ∈L^∞([0,∞);L²(R^d))∩ L²((0,∞); ˙H¹(R^d))be a global weak solution of the initial value problem associated to(2.1)–(2.3). If additionally

(4)

we have V_ij ∈L^∞([t₀,∞);BMO(R^d))for all i, j ∈ {1, . . . , d}and somet₀>0, then there existsα >0such that θ∈C^α([t₀,∞)×R^d).

In analogy with the constructions in [2,11], the proof of Theorem 2.1 consists of two steps. Fort₀>0 fixed, we first prove thatθ∈L^∞([t₀,∞);L^∞(R^d)). The main challenge is to prove the Hölder regularity of the solution, which is achieved by using the method of De Giorgi iteration (cf. [12,14,19]). Note that for divergence-free v∈L²_t,x, the existence of a weak solutionθto (2.1)–(2.3), evolving fromθ₀∈L², is known (for instance, see [27] where the more generalv∈L¹_locis treated, also [2], and references therein). Moreover, this weak solution satisfies the classical energy inequality and the level set energy inequalities (2.6) below.

Remark 2.2.The conclusion of Theorem 2.1 holds if the Laplacian on the right side of (2.1) is replaced by a generic second-order strongly elliptic operator∂i(aij∂j), with bounded measurable coefficients{aij}.

Remark 2.3.We note that the De Giorgi techniques used here to prove Hölder regularity for solutions to (2.1)–(2.3) can also be used to prove Hölder regularity for the problem with a forcing term Son the right side of (2.1). In this case we considerS∈L^r_t,xto be an externally given force, withr >1+d/2 (cf. [19]).

Remark 2.4.In a very recent preprint, Seregin, Silvestre, Šverák, and Zlatoš [28] also use De Giorgi techniques to prove Hölder regularity of solutions to a parabolic equation with drift velocities inL^∞_t BMO⁻_x¹.

Notation.In the following we shall use the classical function spaces:L^p – Lebesgue spaces,BMO– functions with bounded mean oscillation, BMO⁻¹ – derivatives of BMO functions, H˙^s – homogeneous Sobolev spaces, andC^α – Hölder spaces. To emphasize the different integrability in space and time we shall denoteL^p([0,∞);L^q(R^d))by L^p_tL^q_xfor 1p, q∞, and similarly forL^p_tH˙_x¹andL^p_t BMOx. AlsoL^p_t,x(I×B)=L^p(I;L^p(B))for anyI⊂Rand B⊂R^d. The ball inR^dand the parabolic cylinder inR^d⁺²are classically denoted byB_ρ(x₀)= {x∈R^d:|x−x₀|< ρ} andQ_ρ(t₀, x₀)= [t₀−ρ², t₀] ×B_ρ(x₀)forρ >0. Lastly, we shall write(f−k)₊=max{f −k,0}.

2.1. Boundedness of the solution

The first step is to show that a weak solution is bounded for positive time.

Lemma 2.5 (FromL² toL^∞). Let θ∈L^∞([0,∞);L²(R^d))∩L²((0,∞); ˙H¹(R^d)) be a global weak solution of (2.1)–(2.3)evolving fromθ0∈L²(R^d), wherev∈L²((0,∞);L²(R^d)). Then for allt >0we have

θ (t,·)

L^∞(R^d)Cθ₀L²(R^d)

t^d/4 , (2.5)

for some sufficiently large positive dimensional constantC.

Proof. The proof of this lemma is mutatis-mutandis as in [2,11], and requires only the fact thatvis divergence free.

The main idea is that sincey→(y−h)₊is convex, for allh >0 we have

∂_t(θ−h)₊−(θ−h)₊+(v· ∇)(θ−h)₊0,

and hence, multiplying by(θ−h)₊integrating by parts, and using that divv=0, we obtain the energy inequality

R^d

θ (t2,·)−h

+²dx+2

t₂

t₁

R^d

∇(θ−h)₊²dx dt

R^d

θ (t1,·)−h

+²dx, (2.6)

for allh >0 and 0< t1< t2<∞. Fort0>0, andH >0 to be chosen sufficiently large, we definetn=t0−t0/2ⁿ, hn=H−H /2ⁿ, and

c_n=sup

ttn

R^d

θ (t,·)−h_n

+²dx+2 ∞ tn

R^d

∇(θ−h_n)²dx dt,

(5)

wheren0. The inequality (2.6), the Gagliardo–Nirenberg–Sobolev inequality, and Riesz interpolation then imply that

cn+1 C

t₀H^4/d2ⁿ⁽¹⁺^4/d)cn¹⁺^2/d.

LettingH=Cc^1/2₀ /t₀^d/4Cθ0L²(R^d)/t₀^d/4, for some sufficiently large dimensional constantC, implies thatcn→0 exponentially asn→ ∞, and thereforeθ (t0,·)H. Applying the same procedure to−θconcludes the proof of the lemma. We refer the reader to [2,11] for further details. 2

2.2. Local energy and uniform inequalities

In proving the boundedness of the solution we only required thatv∈L²_t,x, and divv=0. For the rest of the section we use the additional assumptionv∈L^∞_t BMO⁻_x¹.

Lemma 2.6(First energy inequality). Letθ∈L^∞_t L²_x∩L²_tH˙_x¹be a global weak solution of the initial value problem associated to(2.1)–(2.3). Furthermore, assume thatVij∈L^∞((0,∞);BMO(R^d))for alli, j∈ {1, . . . , d}, and(2.4) holds. Then for any0< r < Randh∈R, we have

(θ−h)₊²

L^∞_t L²_x(Qr)+∇(θ−h)₊²

L²_t,x(Qr) CR

(R−r)²(θ−h)₊²⁻^d⁺²²

L²_t,x(Q_R)(θ−h)₊^d⁺²²

L^∞_t,x(Q_R), (2.7) whereC=C(d,V_ijL^∞_t BMO_x)is a fixed positive constant, and we have denotedQ_ρ= [t₀−ρ², t₀] ×B_ρ(x₀)for ρ >0and an arbitrary(t₀, x₀)∈(0,∞)×R^d. Moreover, estimate(2.7)also holds withθreplaced by−θ.

Remark 2.7.Note that from Lemma 2.5 we have thatθ∈L^∞_t,x, and hence the right side of (2.7) is finite.

Remark 2.8. The classical local energy inequality (cf. [14,19,25], see also [2,11]) does not contain the term (θ−h)₊L^∞_t,x(QR) on the right, since the velocity fieldvis not as singular as in our case. In this section we prove that since in (2.7) the exponent 2/(d+2)of(θ−h)₊L^∞_t,x(Q_R)is “small enough”, the De Giorgi program may still be carried out to obtain the Hölder regularity of weak solutions.

Proof of Lemma 2.6. Fixh∈Rand let 0< r < Rbe such thatt₀/2−R²>0. Letη(t, x)∈C₀^∞((0,∞)×R^d)be a smooth cutoff function such that

0η1 in(0,∞)×R^d,

η≡1 inQ_r(x₀, t₀), and η≡0 in cl

Q^c_R(x₀, t₀)∩

(t, x): tt₀ ,

|∇η| C

R−r, |∇∇η| C

(R−r)², |∂tη| C

(R−r)² inQR(x0, t0)\Qr(x0, t0),

for some positive dimensional constantC. Definet₁=t₀−R²>0 and lett₂∈ [t₀−r², t₀]be arbitrary. Multiply (2.1) by(θ−h)₊η²and then integrate on[t₁, t₂] ×R^dto obtain

t2

t₁

R^d

∂_t

(θ−h)²₊

η²dx dt−2

t2

t₁

R^d

∂_jj(θ−h)₊(θ−h)₊η²dx dt+

t2

t₁

R^d

∂_iV_ij∂_j

(θ−h)²₊ η²dx dt

=0. (2.8)

The main obstruction to applying classical the de Giorgi estimates (via theL^p-based Caccioppoli inequality, cf. [14, 19]) is that∂_iV_ij ∈L^∞_t BMO⁻_x¹, as opposed to the case L^∞_t W_x⁻^1,^∞ considered by Osada [25] (see also [27]). We overcome this difficulty by subtracting fromV_ij(t,·)its spatial mean over{t} ×B_R, namelyV_ij,B_R(t )(this does not introduce any lower order terms because∂_x_iV_ij,B_R(t )=0), and by appealing to the John–Nirenberg inequality. More precisely, we define

(6)

V_ij,R(t, x)=V_ij(t, x)−V_ij,B_R(t )=V_ij(t, x)− 1

|BR|

BR

V_ij(t, y) dy, (2.9)

and note that∂_iV_ij =∂_iV_ij,R. Therefore, the third term on the left of (2.8) may be replaced by

t2

t1

R^d

∂_iV_ij,R∂_j

(θ−h)²₊

η²dx dt.

We integrate by parts intthe first term on the left of (2.8), and useη(t1,·)≡0. The second term we integrate twice by parts inx_j, and the third term on the left of (2.8) we integrate by parts first inx_j (and use∂_j(∂_iV_ij,R)=∂_i(∂_jV_ij)=

∂_jv_j=0) and then integrate by parts inx_i, to obtain 1

2

R^d

θ (t₂,·)−h2

+η(t₂,·)²dx+

t2

t₁

R^d

∇(θ−h)₊²η²dx dt

=

t2

t1

R^d

(θ−h)²₊η∂_tη dx dt+

t2

t1

R^d

(θ−h)²₊∂_j(η∂_jη) dx dt−

t2

t1

R^d

V_ij,R(θ−h)²₊∂_i(η∂_jη) dx dt

−2

t2

t1

R^d

V_ij,R∂_i(θ−h)₊(θ−h)₊η∂_jη dx dt. (2.10)

Using the bounds on the time and space derivatives ofη, the fact thatη≡1 onQ_r,t₂t₀, the Hölder andε-Young inequalities, we obtain from (2.10)

Br

θ (t₂,·)−h2 +dx+2

t2

t1

R^d

∇(θ−h)₊²η²dx dt

C

(R−r)²

QR

(θ−h)²₊dx dt+ C (R−r)²

QR

|V_ij,R|(θ−h)²₊dx dt

+

t2

t1

R^d

∇(θ−h)₊²η²dx dt+ C (R−r)²

Q_R

|V_ij,R|²(θ−h)²₊dx dt. (2.11)

After absorbing the third term on the right of (2.11) into the left side, we take the supremum overt₂∈ [t₀−r², t₀], to obtain

(θ−h)₊²

L^∞_t L²_x(Qr)+∇(θ−h)₊²

L²_t,x(Qr)

C

(R−r)²(θ−h)₊²

L²_t,x(QR)+ C (R−r)²

Q_R

|Vij,R|(θ−h)²₊dx dt

+ C

(R−r)²

Q_R

|Vij,R|²(θ−h)²₊dx dt. (2.12)

As a corollary of the celebrated John–Nirenberg inequality (cf. [14,30]) we have that for any fixed R >0, t ∈ [t₀−R², t₀], and 1< p <∞,

V_ij,R(t,·)

L^p(B_R)=V_ij(t,·)−V_ij,B_R(t )

L^p(B_R)CV_ij(t,·)

BMO(R^d)|B_R|^1/p,

(7)

whereC=C(d, p) >0 is a fixed constant (recall thatC(d, p)→ ∞asp→ ∞). The fact thatV_ij∈L^∞([t₀/2,∞); BMO(R^d))implies that for allt∈ [t₀−R², t₀]we have

Vij,R(t,·)

L^p(BR)C0|BR|^1/p (2.13)

for a positive constantC0=C0(V_ijL^∞([t₀/2,∞);BMO(R^d)), d, p). We fix 0< ε <2 to be chosen later, and using (2.13) and the Hölder inequality we bound

Q_R

|Vij,R|(θ−h)²₊dx dt=

t0

t0−R² B_R

Vij,R(t, x)(θ−h)²₊(t, x) dx

dt

C₀|B_R|^ε/2

t0

t0−R²

θ (t,·)−h

+²

L^4/(2⁻^ε)(BR)dt.

Using the interpolation inequalityfL^pCf^2/p_L2 f¹_L⁻∞^2/p, withp=4/(2−ε), we obtain from the above esti- mate that

QR

|V_ij,R|(θ−h)²₊dx dtC₀|B_R|^ε/2

t0

t0−R²

θ (t,·)−h

+²⁻^ε

L²(B_R)θ (t,·)−h

+^ε

L∞(B_R)dt

C0R^ε(d⁺^2)/2(θ−h)₊²⁻^ε

L²_t,x(Q_R)(θ−h)₊^ε

L^∞_t,x(Q_R). (2.14)

Similarly, from (2.13), the Hölder inequality andL^pinterpolation, we obtain

Q_R

|Vij,R|²(θ−h)²₊dx dtC0R^ε(d⁺^2)/2(θ−h)₊²⁻^ε

L²_t,x(QR)(θ−h)₊^ε

L^∞_t,x(QR). (2.15)

Combining estimates (2.12) with (2.14), (2.15), and the Hölder inequality, we conclude that (θ−h)₊²

L^∞_t L²_x(Qr)+∇(θ−h)₊²

L²_t,x(Qr)

C0R^ε(d⁺^2)/2

(R−r)² (θ−h)₊²⁻^ε

L²_t,x(QR)(θ−h)₊^ε

L^∞_t,x(QR). (2.16)

The proof of the lemma is concluded by lettingε=2/(d+2)in (2.16). 2 By applying the Hölder inequality to the right side of (2.7) we then obtain:

Corollary 2.9.Letθbe as in Lemma2.6. Then we have (θ−h)₊²

L^∞_t L²_x(Qr)+∇(θ−h)₊²

L²_t,x(Qr) CR^d⁺²

(R−r)²(θ−h)₊²

L^∞_t,x(QR), (2.17)

for some positive constantC=C(d,V_ijL^∞_t BMO_x).

We now fix a point(t₀, x₀)∈(0,∞)×R^dand we prove the Hölder continuity ofθat this point. Throughout the following we denote byQ_ρthe cylinderQ_ρ(t₀, x₀), for anyρ >0.

The following lemma gives an estimate on the supremum ofθon a half cylinder, in terms of the supremum on the full cylinder. A similar statement may be proven for−θ.

Lemma 2.10.Letθbe as in Lemma2.6. Assume thath₀sup_Q_r

0θ, wherer₀>0is arbitrary. We have sup

Q_r

0/2

θh₀+C

|{θ > h₀} ∩Q_r₀|^1/(d⁺²⁾ r₀

1/2 sup

Q_r

0

θ−h₀

(2.18) for some positive constantC=C(d,V_ijL^∞_t BMO_x).

(8)

The above estimate differs from the classical one cf. [19, Theorem 6.50] in that the power of|{θ > h₀}∩Q_r₀|/|Q_r₀| is 1/(2d+4)instead of 1/(d+2). However, the key feature of (2.18) is that the coefficient of (sup_Q_r

0θ−h₀) does not scale withr₀. It is convenient to introduce the following notation:

• A(h, r)= {θ > h} ∩Q_r,

• a(h, r)= |A(h, r)|,

• b(h, r)= (θ−h)₊²_L2 t,x(Q_r),

• M(r)=sup_Q_rθ,

• m(r)=infQrθ,

• osc(Q)=sup_Qθ−infQθ.

Proof of Lemma 2.10. Let 0< r < Rand 0< h < H. We have b(h, r)= θ−h²_L2

t,x(A(h,r))θ−h²_L2

t,x(A(H,r))(H−h)²a(H, r). (2.19)

Let η(t, x)∈C₀^∞(R×R^d)be a smooth cutoff such thatη≡1 onQ_r,η≡0 onQ^c_(r₊_R)/2∩ {t t₀}, and |∇η| C/(R−r)for some universal constantC >0. Then, by Hölder’s inequality and the choice ofηwe obtain

b(h, r)=(θ−h)₊²

L²_t,x(Q_r)

a(h, r)^2/(d⁺²⁾(θ−h)₊²

L^2(d_t,x⁺^2)/d(Q_r)

a(h, r)^2/(d⁺²⁾η(θ−h)₊²

L^2(d_t,x⁺^2)/d((−∞,t0)×R^d). (2.20)

Using the Gagliardo–Nirenberg–Sobolev inequality and Riesz interpolation f²_L2(d+2)/d((−∞,t₀)×R^d)Cf²_L∞

t L²_x((−∞,t₀)×R^d)+C∇f²_L2

t,x((−∞,t₀)×R^d), estimate (2.20) implies that

b(h, r)Ca(h, r)^2/(d⁺²⁾η(θ−h)₊²

L^∞_t L²_x((−∞,t₀)×R^d)+∇

η(θ−h)₊²

L²_t,x((−∞,t₀)×R^d)

Ca(h, r)^2/(d⁺²⁾(θ−h)₊²

L^∞_t L²_x(Q_(r₊_R)/2)+∇(θ−h)₊²

L²_t,x(Q_(r₊_R)/2)

+ 1

(R−r)²(θ−h)₊²

L²_t,x(Q_(r₊_R)/2)

for some positive dimensional constantC. Using the first energy inequality, i.e., Lemma 2.6, and the Hölder inequality, we bound the far right side of the above and obtain

b(h, r)Ca(h, r)^2/(d⁺²⁾ R

(R−r)²(θ−h)₊²⁻^2/(d⁺²⁾

L²_t,x(Q_R) (θ−h)₊^2/(d⁺²⁾

L^∞_t,x(Q_R)

Ca(h, r)^2/(d⁺²⁾ R

(R−r)²b(h, R)¹⁻^1/(d⁺²⁾(θ−h)₊^2/(d⁺²⁾

L^∞_t,x(Q_R) (2.21)

for some sufficiently large positive constantC=C(d,V_ijL^∞_t BMO_x). By combining estimates (2.19) and (2.21) we obtain the main consequence of Lemma 2.6, that is

b(H, r) CR

(H−h)^4/(d⁺²⁾(R−r)²b(h, R)¹⁺^1/(d⁺²⁾(θ−H )₊^2/(d⁺²⁾

L^∞_t,x(Q_R). (2.22)

The above estimates give the proof of the lemma as follows. Let r_n =r0/2+r0/2ⁿ⁺¹r0/2, h_n =h_∞− (h_∞−h0)/2ⁿh_∞, and b_n=b(h_n, r_n₊1), for all n0, where r0 andh0 are as in the statement of the lemma, whileh_∞>0 is to be chosen later. By lettingH=h_n₊₁,h=h_n,r=r_n₊₂, andR=r_n₊₁in (2.22), we obtain

(9)

b_n₊₁ Cr_n₊₁

(h_∞−h0)^4/(d⁺²⁾r₀²2ⁿ⁽²⁺^4/(d⁺²⁾⁾b¹_n⁺^1/(d⁺²⁾(θ−h_n₊₁)₊^2/(d⁺²⁾

L^∞_t,x(Q_rn₊₁)

C(M(r₀)−h₀)^2/(d⁺²⁾

(h_∞−h₀)^4/(d⁺²⁾r₀ 2ⁿ⁽²⁺^4/(d⁺²⁾⁾b¹_n⁺^1/(d⁺²⁾, (2.23) by using

(θ−h_n₊₁)₊

L^∞_t,x(Q_rn+₁)= sup

A(h_n+1,r_n+1)

θ−h_n₊₁ sup

Q_rn₊₁

θ−h_n₊₁sup

Qr0

θ−h₀=M(r₀)−h₀ which holds sinceM(r_n)M(r₀)andh_nh₀. LetB=2⁴⁺^2(d⁺²⁾. We chooseh_∞large enough so that

C(M(r₀)−h₀)^2/(d⁺²⁾

(h_∞−h₀)^4/(d⁺²⁾r₀ b^1/(d₀ ⁺²⁾ 1

B, (2.24)

then by induction we obtain from (2.23) that b_n b₀/Bⁿ, and therefore b_n→0 as n→ ∞. This implies that sup_Q_r

0/2θh_∞. A simple calculation shows that if we let h_∞=h₀+CB^(d⁺^2)/4(M(r0)−h0)^1/2b^1/4₀

r₀^(d⁺^2)/4 (2.25)

then (2.24) holds. Lastly,b₀=b(h₀,3r0/4)may be bounded via (2.21) and the Hölder inequality as b₀Ca(h₀, r₀)^1/(d⁺²⁾r₀^d⁺²

M(r₀)−h₀2

. (2.26)

The proof of the lemma is concluded by combining sup_Q_r

0/2θh_∞with (2.25) and (2.26). From the above proof it follows that inequality (2.18) also holds withθis replaced by−θ. 2

As opposed to the elliptic case, in the parabolic theory we need an additional energy inequality to control the possible growth of level sets of the solution.

Lemma 2.11(Second energy inequality). Letθ∈L^∞_t L²_x∩L²_tH˙_x¹be a global weak solution of the initial value problem associated to(2.1)–(2.3). Furthermore, assume thatV_ij ∈L^∞_t BMOx for alli, j∈ {1, . . . , d}, and(2.4)holds. Fix an arbitraryx₀∈R^d, leth∈R,0< r < R, and0< t₁< t₂. Then we have

θ (t2,·)−h

+²

L²(Br)θ (t1,·)−h

+²

L²(B_R)+C R^d(t₂−t₁)

(R−r)² (θ−h)₊²

L^∞_t,x((t₁,t₂)×B_R) (2.27) for some sufficiently large positive constantC=C(d,VijL^∞_t BMOx), where we have denotedBρ=Bρ(x0)forρ >0.

Proof. Note that by Lemma 2.5 we have that θ∈L^∞_t,x and hence the right side of (2.27) is finite. Letη∈C₀^∞(R^d) be a smooth cutoff such thatη≡1 onB_r,η≡0 onB_R^c, and|∇η(x)|C/(R−r), for allx∈R^d, for some constant C >0. Multiply (2.1) byη²(θ−h)₊and integrate fromt₁tot₂to obtain

t₂

t1

R^d

∂t

(θ−h)₊2

η²dx dt−2

t₂

t1

R^d

∂jj(θ−h)₊(θ−h)₊η²dx dt

= −

t₂

t₁

R^d

∂_iV_ij∂_j

(θ−h)₊2

η²dx dt

= −

t2

t1

R^d

∂_iV_ij,R∂_j

(θ−h)₊2

η²dx dt,

where, as in (2.9), we have denotedV_ij,R(t, x)=V_ij(t, x)−_|_B¹_R_|

B_RV_ij(t, y) dy. After integrating by parts we get