On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation

We consider a transport-diffusion equation of the form $\partial_t \theta +v \cdot \nabla \theta + \nu \A \theta =0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\A$ represents log-modulated fractional dissipation: $\A=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator $\A$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $ |\theta(t)|_\infty \le e^{Ct} |\theta_0|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $|\theta(t)|_p$ with $1\le p \le \infty$. At the cost of an exponential factor, this extends a recent result of Hmidi (2011) to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.

The main objective of this paper is to prove some maximum principles for the operator A and A 1 in Lebesgue spaces.
The transport-diffusion model (1.1) is a natural generalization of several linear and nonlinear fluid equations such as the two-dimensional surface quasi-geostrophic equations, fractional Burgers equations, vortex patch models, and Boussinesq systems. See, for instance, the recent work [9,4,3,10,6,5] and references therein. In these problems, the velocity v is typically related to the active scalar θ by a constitutive relation v = T (θ) where T could be some singular integral operator or more generally a nonlocal operator. To obtain local and global wellposedness results for the nonlinear problems, an important first step is to get a priori L p , 1 ≤ p ≤ ∞ estimates of solutions. Specific to the linear problem (1.1), one needs to prove L p bounds on the active scalar θ independent of the size of v. We refer to these types of results as L p maximum principle estimates. In this respect, the two-dimensional dissipative surface quasi-geostrophic equations can be regarded as a (nonlinear) version of (1.1) and they correspond to the case β = 0 in the operator A. A classical result is due to A. Córdoba and D. Córdoba [2] who proved the following In a recent article, Hmidi [7] initiated the study of (1.1) and obtained the following important maximum principle: Then any smooth solution of (1.1) satisfies To prove Theorem 1.1, Hmidi used the theory of C 0 -semigroup of contractions on L p (1 < p < ∞) for the family of convolution kernels (K t ) t≥0 defined by The key step is to get the positivity of the kernel K t . For this purpose, Hmidi used the Askey's criterion for characteristic functions [1]. The restrictions on the dimension d and the parameters (γ, β, λ) are mainly due to the use of this criterion. Hmidi conjectured that the maximum principle should hold for all dimensions d ≥ 1 and the full range 0 ≤ γ ≤ 2 and β ≥ 0. The purpose of this paper is to give an affirmative answer to this question at the cost of a harmless exponential factor. Theorem 1.2 (Generalized maximum principle, L ∞ case). Let ν ≥ 0, d ≥ 1, 0 ≤ γ ≤ 2 and β ≥ 0, λ > 1. Assume θ = θ(t, x) is a smooth solution of (1.1) which decays at spatial infinity, i.e., for any fixed t ≥ 0, Then we have where C > 0 is a constant depending only on (ν, d, γ, β, λ).
Remark 1.3. The same result holds if we replace the dissipation operator A by A 1 in (1.1). As we shall see in Section 2, the proof for A 1 case is actually simpler. The decay condition (1.2) is fairly weak as most smooth solutions to these type of fluid equations typically belong to the Sobolev space C 0 t H s x which can easily imply (1.2). We should also stress that we do not assume any divergence-free condition on v in Theorem 1.2. This can have applications for compressible fluid equations.
To prove Theorem 1.2, we shall use a completely new idea which avoids the use of Askey's criterion. Namely we introduce a novel nonlocal decomposition of the operator A (see Section 2 for more details) in terms of a weighted integral of the usual fractional operators |∇| s , 0 ≤ s ≤ γ plus a smooth remainder term which corresponds to an L 1 kernel. Thanks to this new decomposition, we shall only need to appeal to the classic maximum principle for the fractional Laplacian operators. In a similar vein, one can even consider a weighted integral of a parameterized family of nonlocal operators each of which obeys a maximum principle. However we shall not pursue this generality here.
As was already mentioned, Theorem 1.2 deals with the L ∞ norm and no special assumption is needed on the velocity field v. On the other hand for more general L p -norms with 1 ≤ p < ∞, the divergence-free condition on the vector field v has to be assumed, as one needs to calculate the time derivative of the L p norm and perform integration by parts.
where C > 0 is a constant depending only on (ν, d, γ, β, λ). It is an interesting question whether one can prove the sharp constant is C = 0 in both Theorem 1.2 and Theorem 1.4. We conjecture this is indeed the case at least for a generic set of parameters.
The rest of this article is organized as follows. In Section 2 we introduce the nonlocal decomposition for both the operator A and the operator A 1 . In Section 3 we give the proof for Theorem 1.2 and Theorem 1.4 for the operator A 1 . The case 0 ≤ γ ≤ 1 of A is also covered there. In Section 4 we complete the proof of the main theorems for the operator A in the regime 1 < γ ≤ 2.
We conclude the introduction by setting up some Notations.
• For any two quantities X and • For any f on R d , we denote the Fourier transform of f has The inverse Fourier transform of any g is given by 1 We thank Edriss Titi for this suggestion.
• For any real number x, the sign function sgn(x) is defined as follows For any complex z with Re(z) > 0, the Gamma function Γ(z) is given by the expression • We will also occasionally need to use the Littlewood-Paley frequency projection operators. Let ϕ(ξ) be a smooth bump function supported in the ball |ξ| ≤ 2 and equal to one on the ball |ξ| ≤ 1. For each dyadic number N ∈ 2 Z we define the Littlewood-Paley operators , , Similarly we can define P <N , P ≥N , and P M<·≤N := P ≤N − P ≤M , whenever M and N are dyadic numbers.
Acknowledgements. H. Dong was partially supported by the NSF under agreements DMS-0800129 and DMS-1056737.

The nonlocal decomposition
We start with the following lemma which establishes the nonlocal decomposition of the log-modulated fractional dissipation operator |∇| γ log β (λ+|∇|) in the regime 0 ≤ γ ≤ 1. One should notice the subtle difference between this operator and the operator A 1 in the logarithmic term. By a simple change of variable |ξ| → |ξ| 2 , the decomposition of the operator A 1 is addressed in the next corollary. After that we establish the decomposition for the operator A in the regime 1 < γ ≤ 2. The proof will be more involved due to certain first order negative corrections.
where P is a smooth Fourier multiplier which maps L p to L p for all 1 ≤ p ≤ +∞. More precisely, for any function f To show (2.2) we start with the simple identity We then set C β = 1 Γ(β) and obtain (2.2) with It remains for us to show the L 1 boundedness of F −1 (P ). We first deal with the piece (2.4). By the Fundamental Theorem of Calculus, we have for any 0 ≤ t ≤ 1, If 0 < t < 1, then Since for y > 0 the Poisson kernel F −1 (e −y|ξ| ) is positive, it follows easily that F −1 ((s + |ξ|) t−1 ) is a non-negative function and furthermore, Plugging the above estimate into (2.4), we get which is clearly good for us. For (2.5), we just note that for τ > γ, where we used the fact that λ > 1.
Finally we deal with the contribution of (2.6). By (2.3), it is obvious that since λ > 1. Now in (2.6) we may assume 0 < γ < 1 (the cases γ = 0 and γ = 1 are trivial). By (2.8) and Young's inequality, we get By a simple substitution |ξ| → |ξ| 2 , we can deduce the nonlocal decomposition of the operator A 1 from Lemma 2.1. Of course, one still needs to check the L 1 boundedness of the error term under such nonlinear substitution.

9)
where P is a smooth Fourier multiplier which maps L p to L p for all 1 ≤ p ≤ +∞. More precisely, for any function f and K L 1 x ≤ Const. Proof of Corollary 2.2. On the Fourier side, the identity (2.9) is equivalent to the following (the value of the constant C β can be adjusted slightly) By using a similar derivation as in the beginning part of the proof of Lemma 2.1 (see in particular (2.3)-(2.6), and replace |ξ| by |ξ| 2 , γ by γ/2), we obtain (2.10) with Note that 0 ≤ γ 2 ≤ 1 and the fact F −1 (a + |ξ| 2 ) −s L 1 x = a −s for any s > 0 and a > 0. By using a similar analysis as in the proof of Lemma 2.1, it is then not difficult to check that F −1 (P ) is an L 1 bounded kernel.
We now consider the more involved 1 < γ ≤ 2 case for the operator A. One should compare the decomposition (2.11) with (2.1). Lemma 2.3 (Nonlocal decomposition, case 1 < γ ≤ 2). Let d ≥ 1, 1 < γ ≤ 2 and β > 0, λ > 1. Then we have the decomposition: where P is a smooth Fourier multiplier which maps L p to L p for all 1 ≤ p ≤ +∞. More precisely, for any function f Lemma 2.3. Throughout this proof we shall use the letter P to denote the symbol of an L 1 → L 1 bounded operator. For the convenience of notation, we allow the value of P to vary from line to line. We begin with two elementary estimates. Let φ ∈ C ∞ c (R d ) be a radial smooth cut-off function such that φ(x) = 1 for |x| ≤ 2 and φ(x) = 0 for |x| ≥ 3. For any constant C > 0 define φ <C (x) := φ(x/C) and φ >C (x) = 1 − φ <C (x). Then for any γ > 0, C > 0, s ≥ 0, we have the following where the constant C 1 is independent of s. To prove (2.12), one can use a scaling argument to reduce to the λ = 1 case. The result then follows easily from the binomial expansion of (|ξ|/(1 + |ξ|)) γ = (1 − 1 1+|ξ| ) γ = n≥0 C γ,n (1 + |ξ|) −n , the L 1 -boundedness of the operators (1 + |ξ|) −n (namely F −1 ((1 + |ξ|) −n ) L 1 x →L 1 x ≤ 1 for any n ≥ 0), and the fact that n≥0 |C γ,n | < +∞ (note that C n,γ has a definite sign for n sufficiently large). To prove (2.13) one can again use scaling to reduce to the case C = 1. For s ≥ 2 the result is obvious by using integration by parts. For 0 < s ≤ 2, we note that and on the support of φ >C (ξ), One can then use the Littlewood-Paley operators to bound (since we are summing over N ≥ N 0 the convergence in L 1 is of no problem): Observe that the implied constants are uniform in s since 0 < s ≤ 2. This settles (2.13).

By (2.3), we write
We first deal with (2.16). Rewrite By using (2.12) we obtain Next we turn to (2.15). By inserting a smooth cut-off function φ >10λ (ξ), We have On the other hand, by using the binomial expansion of the function (1 + t) −s = n≥0 C n,s t n and the estimate (2.13), it is not difficult to check that Hence we have proved We turn now to the final piece (2.14). The main idea is similar to that of (2.15). We again insert a smooth cut-off φ >10λ (ξ) and observe that when |ξ| ≥ 10λ, we have Note that 0 ≤ τ ≤ γ − 1 and we have to keep terms up to the linear term. Then clearly The desired decomposition (2.11) now follows.

Proof of Theorem 1.2 and Theorem 1.4 for the operator A 1
In this section we give the proofs of Theorems 1.2 and 1.4 for the operator A 1 . The proofs for the operator A is slightly more involved and will be given in the next section.
Proof of Theorem 1.2 for the operator A 1 . Assume first 0 < γ ≤ 2 and β > 0. By (2.9) we can write where L = C β γ 0 τ β−1 |∇| γ−τ dτ and P L 1 x < +∞. Now take λ 1 > ν P L 1 x and define f (t, x) = e −λ1t θ(t, x). Fix T > 0 and consider Without loss of generality we can assume By using the decay condition (1.2) and a simple compactness argument in t, we conclude that there exists (t 0 , x 0 ) such that We now show that t 0 = 0. Indeed if 0 < t 0 ≤ T , we compute Now by Corollary 2.2, we have For any 0 < s < 2, by using the fractional representation it is easy to see (|∇| s f )(t 0 , x 0 ) ≥ 0 and hence

Plugging (3.3) and (3.4) into (3.2)
, we reach a contradiction: Therefore we conclude that t 0 = 0 and clearly the estimate (1.3) follows. It remains to prove the case γ = 0 and β > 0. But in this case the operator log −β (λ − ∆) corresponds to an L 1 -bounded convolution kernel. Hence we just need to repeat the previous argument with A 1 = P and λ 1 > ν P L 1 x . We omit the repetitive details.
Finally we complete the Proof of Theorem 1.4 for the operator A 1 . Without loss of generality we assume ν > 0, 0 < γ ≤ 2 and β > 0. Let 1 ≤ p < ∞. Multiplying both sides of (1.1) by |θ| p−1 sgn(θ), integrating by parts and using the fact that v is divergence free, we where in the last equality we have used the decomposition (3.1).
Since for any 0 ≤ s < 2, 1 ≤ p < ∞, we have By Hölder, we have x . Plugging the above estimates into (3.5) and integrating in time, we get for any 1 ≤ p < ∞, The case p = ∞ follows by a limiting argument p → ∞. Clearly (1.4) holds by setting C = ν P L 1 x . 4. Proof of the main theorems for the operator A In this section we describe the proofs of the main theorems for the operator A. We shall only need to consider the case 1 < γ ≤ 2. Thanks to Lemma 2.1 the case 0 ≤ γ ≤ 1 is already covered in the previous section. In the case 1 < γ ≤ 2 we have to use the decomposition (2.11) in Lemma 2.3. The extra complication is due to the negative term −λτ |∇| γ−τ −1 which in principle can cause the maximum principle to fail. The way out of this difficulty is to note that the main term |∇| γ−τ is stronger than this negative term by an order of |∇| −1 . The following lemma quantifies this observation. In some sense it gives the maximum principle for "mixed" operators.
Then for any smooth function g which attains its maximum at some point x 0 , we have (4.1) where C d > 0 is some constant depending only on the dimension d. In particular if s 1 = s 2 − 1 and 1 < s 2 < 2, then we have the estimate

2)
where C ′ d > 0 is another constant depending only on the dimension d.
Proof of Lemma 4.1. To begin we need to derive the explicit constant appearing in the integral representation of the fractional operators |∇| s with 0 < s < 2.
Recall that for any 0 < α < d, the Riesz potential |∇| −α has the following explicit representation (cf. pp 117 of Stein [8]) For 0 < s < 2 by writing |∇| s = −∆|∇| −(2−s) and integrating by parts, we get By using the asymptotics Γ(z) ∼ z −1 for z ∼ 0, it is easy to see that and in particular for all 0 < s < 2, We now write Observe that g(x 0 ) − g(y) ≥ 0 for all y. We now separate the y-integral into two regimes. The first regime is {y : |y − x 0 | ≤ min{1, C 2 }}, where C 2 > 0 is a constant such that (here we use (4.4) to bound C s1,d ) By using (4.3), we have (the notation ∼ C1,d means up a constant depending on C 1 and d) In the first regime, it is easy to check that the first integral bounds the second integral in (4.5). The second regime is just the complement {y : |y − x 0 | > min{1, C 2 }}. In this case we simply discard the first integral and bound the second integral by g L ∞ x which produces a term of the form (below C d denotes a constant depending only on the dimension d): This settles (4.1).
The following corollary will be used in the proof of Theorem 1.2.
Proof of Corollary 4.2. By Lemma 2.3, Lemma 4.1, and the fact that (|∇| γ−τ g)(x 0 ) ≥ 0, we have We are now ready to complete the Proof of Theorem 1.2 for the operator A, case 1 < γ ≤ 2. With the help of Corollary 4.2, the proof is similar to the proof for the operator A 1 in section 3, one only needs to consider f (t, x) = e −λ1t θ(t, x) with λ 1 > ν(C d,β,γ λ + P L 1 x ), where the constant C d,β,γ is the same as in Corollary 4.2. The rest of the proof is now the same as in Section 3. We omit the details.
Next we turn to the proof of Theorem 1.4. The following lemma establishes a form of maximum principle for the mixed operator L = |∇| s − C 1 |∇| s−1 in the L p , 1 ≤ p < ∞ setting. Then for any 1 ≤ p < ∞ and any smooth g ∈ L p , we have the bound R d (Lg)|g| p−1 sgn(g) dx ≥ −C 1 C d,γ g p p , (4.6) where C d,γ > 0 is some constant depending only on (d, γ). and (4.9) + (4.10) ≥ 0.
Substituting the value of A into (4.12), we obtain (4.6).
Finally we are ready to complete the Proof of Theorem 1.4 for the operator A in the regime 1 < γ ≤ 2. Thanks to Lemma 2.3 and Lemma 4.3, we essentially only have to repeat the proof for the operator A 1 in Section 3. In place of (3.5), we have 1 p d dt ( θ(t) p p ) ≤ ν( P L 1 x + λC d,γ,β ) θ(t) p p .