Ying Hu
email: ying.hu@univ-rennes1.fr.

Zhongmin Qian
email: qianz@maths.ox.ac.uk.

Zichen Zhang
email: zhangz@maths.ox.ac.uk.

Gradient estimates for porous medium and fast diffusion equations by martingale method

Keywords: Negative finite variation part

estimate for solutions to the heat equation 10 3 Gradient estimate for positive solutions to PME and FDE 13 3.1 Local gradient estimate . . . . . . . .

Introduction

The present paper is devoted to a study of gradient estimates for solutions of partial differential equations (PDEs) of the form

∂u ∂t = ∆u m (1)
where m ∈ (0, ∞) is a given exponent. The problem is posed on (0, ∞) × R n , and ∆ is the Laplace operator with respect to space variables. When m = 1, it is the celebrated heat equation, which is linear and parabolic, hence enjoying many nice properties. When m = 1, the story becomes quite different. Let us focus on non-negative solutions to the equation [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF] with initial condition u (0, x) = u 0 (x) , the so-called Cauchy problem related to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF], and collect some results and discuss some phenomena that do not appear in the case of the heat equation.

There are two critical values of the exponent m, namely m = 1 and m = n-2 n . By rewriting [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF] as

∂u ∂t = mu m-1 ∆u + m (m -1) u m-2 |∇u| 2 (2)
one can see that the coefficient matrix of the second order derivative is mu m-1 I n×n . If m < 1, this matrix takes the value of infinity where u = 0, which means the parabolicity is singular. While if m > 1, the matrix vanishes when u = 0, which means the parabolicity is degenerate. If we interpret (1) as a differential equation describing a diffusion, this means when m < 1, the diffusion is very fast at places where u is small. Therefore in this case we call (1) the fast diffusion equation (FDE). If m > 1, the diffusion slows down wherever u is small, and in this case we call it the porous medium equation (PME). Notice that since we only consider non-negative solution u, from (2) it is clear that (1) is always formally parabolic. Due to the different behaviors of PME and FDE at u = 0, theories about existence and uniqueness of Cauchy problems for these two types of equations have been studied separately. The commonly used framework for PME is the L 1 (R n ) space. By Theorem 9.12 and Proposition 9.13 in [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF], if u 0 ∈ L 1 (R n ), then there is a unique strong solution which is continuous on (0, ∞) × R n . Moreover, if u 0 ∈ L 1 (R n ) which is strictly positive and continuous, then the solution must be smooth. If we move beyond the scope of L 1 (R n ) setting and impose a weaker growth condition on u 0 such as

sup R≥1 R -(n+ 2 m-1 ) ˆ|x|≤R |u 0 (x)| dx < ∞, (3)
then by [START_REF] Bénilan | Solutions of the porous medium equation in R N under optimal conditions on initial values[END_REF] a unique solution in distribution sense exists on (0, T (u 0 )) × R n , where T (u 0 ) ∈ (0, ∞] depending on u 0 . By Theorem 3.1 in [START_REF] Aronson | The initial trace of a solution of the porous medium equation[END_REF], the previous growth condition is satisfied by any continuous non-negative solution. Therefore, condition (3) is optimal for the class of continuous non-negative solutions. If the initial data is allowed to be measure-valued, [START_REF] Dahlberg | Nonnegative solutions of generalized porous medium equations[END_REF] gives a result which requires similar growth condition as [START_REF] Aronson | Régularité des solutions de l'équation des milieux poreux dans R N[END_REF]. As for FDE, no requirement on the growth of initial data is needed. In fact, by Theorem 2.1 and 2.3 in [START_REF] Herrero | The Cauchy problem for u t = ∆u m when 0 < m < 1[END_REF], there exists a unique solution u ∈ C [0, ∞) ;

L 1 loc (R n ) in distribution sense if u 0 ∈ L 1 loc (R n ).
Alternatively, if we impose some growth and decay conditions on u 0 , by Theorem 1 in [START_REF] Bernard | Existence theorems for fast diffusion equations[END_REF] there will be a classical solution in 0, T × R n , and T is finite.

The degeneracy of parabolicity of PME leads to finite propagation of its solution, which is one of the special feature of PME. In particular, by Theorem 14.6 in [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF], if u (t 1 , •) is compactly supported in R n , so is u (t 2 , •) for any t 2 > t 1 . Consequently, for this kind of solutions, there is a set in (0, ∞) × R n that separates the region on which u is positive and the region where u is zero. According to Theorem 3.3 in [START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF], this set, or the so-called free boundary, is locally Hölder continuous on (0, ∞) × R n . Moreover, in viewing it as a family of boundaries in R n indexed by t ∈ (0, ∞), those boundaries expands to infinity as t → ∞ [START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF]. When the solution overflows the support of the initial data at some finite time t ⋆ , we see an improvement in the regularity of the free boundary. More precisely, Theorem 3 in [START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF] asserts that the free boundary is locally Lipschitz continuous on (t ⋆ , ∞) × R n .

Although m = 1 is a crucial value when talking about finite propagation and existence theories of equation [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF], it is not a significant value in the study of extinction in finite time and smoothing effect, where the value m = n-2 n becomes decisive. The extinction in finite time of a solution is the phenomenon that arises only when m < n-2 n . On page 174 in [START_REF] Bénilan | The continuous dependence on ϕ of solutions of u t -∆ϕ(u) = 0[END_REF], it is proved that any solution with initial value

u 0 ∈ L p⋆ (R n ) ∩ L 1 (R n ), where p ⋆ = n(1-m)

2

, becomes identically zero after a finite time. More generally, by Theorem 5.2 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF], the same result holds for u 0 ∈ M p⋆ , where

M p⋆ = f ∈ L 1 loc (R n ) : ˆK |f (x)| dx ≤ C |K| 1-1 p⋆ , ∀K with |K| < ∞ .
According to Lemma 5.6 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF], this is already very close to the sufficient condition for a solution to extinct in finite time. One can see from these results that even a positive initial data may produce a solution that vanishes completely in finite time, which is quite striking. The reason behind, is the failure of conservation of mass when m < n-2 n , as explained in Section 5.5 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF]. Next, let us describe regularities of solutions in terms of boundedness, positivity and smoothness. In general, for any m ∈ (0, ∞) and p ∈ [1, ∞], the solution decreases in L p (R n ) norm as it evolves in time, according to Theorem 7.2 in [START_REF] Vázquez | Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations[END_REF]. Moreover, if m > n-2 n , then by Section 3.4 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF], for any p and q such that 1 ≤ p ≤ q ≤ ∞,

u (t, •) q ≤ c (m, n, p, q) u 0 σ p t -α (4)
where

α = n (q -p) q (n (m -1) + 2p) , σ = p (n (m -1) + 2q) q (n (m -1) + 2p) .
In particular, from this result we see that initial data in

L 1 (R n ) produce so- lutions u (t, •) ∈ L ∞ (R n )
for any time t > 0, which is termed as smoothing effect. When m = n-2 n , this is no longer true. Appendix A.3 in [START_REF] Chasseigne | Theory of extended solutions for fastdiffusion equations in optimal classes of data. Radiation from singularities[END_REF] constructed a solution which is not bounded at any time while still having an initial value in L 1 (R n ). Moreover, when m < n-2 n and n ≥ 3, Theorem 5.14 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF] shows that (4) holds for q = 1 and 1 < p < p ⋆ . This means that L p (R n ) data yield solutions only in L 1 (R n ), a somehow backward smoothing effect. Nevertheless, we are still able to get a bounded solution in the case of m < n-2 n if the initial data belongs to a better space. By Theorem 6.7 in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF]

, if m < n-2 n , p ≥ p ⋆ and u 0 ∈ L p (R n ) + L ∞ (R n ), then u (t,
•) is locally bounded and smooth for any t > 0. Quantitatively, Theorem 2.1 in [START_REF] Bonforte | Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations[END_REF] gives a local upper bound for u (t, •) in terms of the L p (R n ) (p > p ⋆ ) norm of u 0 over a larger local region.

As for positivity and smoothness, Theorem 3.1 in [START_REF] Aronson | The initial trace of a solution of the porous medium equation[END_REF], Theorem 1.1 in [8] and Theorem 1.1 in [START_REF] Bonforte | Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations[END_REF] give local lower bounds for solutions to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]

in terms of local L 1 (R n ) norm of initial data in the cases of m > 1, m ∈ n-2
n , 1 and m ∈ 0, n-2 n respectively. These lead to results about the positivity of solutions. For example, Proposition 1.1 in [START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF] gave a necessary and sufficient condition for the positivity of u (t, x) when m > 1 which read as

sup R>0 R -(n+ 2 m-1 ) ˆ|y-x|≤R |u 0 (y)| dy = ∞.
Besides positivity, if the solution is also locally bounded, then standard quasilinear theory [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type. Izdat[END_REF] implies the smoothness of the solution on that region. In particular, when m ∈ n-2 n , 1 , non-negative locally integrable initial data always produce positive and smooth global solutions, according to the remark after Theorem 2.3 in [START_REF] Herrero | The Cauchy problem for u t = ∆u m when 0 < m < 1[END_REF]. This is not true when m ∈ 0, n-2 n , as non-negative locally integrable initial data is not enough for local boundedness of solutions, which can be seen from solution (0.2) in [START_REF] Bonforte | Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations[END_REF]. Moreover, when m is in this range, extinction in finite time may occur, which kills positivity of solutions in a global scale. As for PME, in general, solutions are only locally Hölder continuous. Theorem 4.1 in [START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF] states that when u 0 is non-negative, bounded and belongs to L 2 (R n ), u m is uniformly Hölder continuous in every set (η 0 , ∞) × R n , η 0 > 0. Theorem 7.17 in [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF] tells us that u is locally Hölder continuous on the region where u is bounded.

During the development of the above works, one of the main tools is the comparison principle for equation [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF], which is established in Theorem 7.3 in [START_REF] Vázquez | Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations[END_REF]. Generally speaking, the comparison is in terms of mass concentration of radially symmetric functions. Once this comparison of mass concentration is obtained, comparison in terms of L p (R n ) (p ∈ [1, ∞]) norm follows. There are several special explicit solutions to (1) that are often used in combination with the comparison principle. Let us only mention three of them here. The first one is source-type solution, also known as Zel'dovich-Kompanyeets-Barenblatt (ZKB) solution, which takes the form

U (t, x) = t -α c -k |x| 2 t -2β 1 m-1 + , with α = n n (m -1) + 2 , β = α n , k = α (m -1) 2mn
and a positive constant c. For m > n-2 n , lim t→0 U (t, x) = C (c, n, m) δ 0 (x)
in the sense of measures. So it is the solution to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF] with Dirac delta as initial trace. One can see that it reproduces the heat kernel as m → 1. Since ZKB solution has compact support when m > 1, it plays an important role in the study of finite propagation speed. When m ≤ n-2 n , U (t, x) is no longer integrable in space variables. Therefore, although still solving (1), it is not a solution to any Cauchy problem related to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]. The second type of solution is a family of self similar solutions, which has the form

U (t, x) = t -λ1 F |x| t -λ2 .
By a scaling argument, it is shown in section 3.2.1 of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type[END_REF] that any solution to (1) on (0, ∞) × R n with initial data |x| -λ3 must have the above form. Notice that in fact the ZKB solution also belongs to this type. The third type of solutions we would like to mention is a variance of the self similar solutions. It has the form

U (t, x) = (T -t) -λ1 F |x| (T -t) -λ2
with T > 0. Usually it satisfies (1) only for t < T , after which it blows up or vanishes. For this type of solutions, probably the most popular and explicit one is

U (x, t; T ) = 2 n - 2 1 -m T -t |x| 2 1 1-m with m < n-2
n , which is a good example of solutions that extinct in finite time. As frequently seen in the field of PDE, a crucial step in the study of equation ( 1) is to derive various types of estimates for solutions. In [START_REF] Aronson | Régularité des solutions de l'équation des milieux poreux dans R N[END_REF], Aronson and Bénilan established the following gradient estimate for solutions to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]. If m ∈ n-2 n , ∞ , u is a positive smooth solution to [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]

and v = m m-1 u m-1 , then ∆v ≥ - α (m -1) t where α = n(m-1)
n(m-1)+2 , which is equivalent to

|∇v| 2 (m -1) v - v t (m -1) v ≤ α (m -1) t .
This fundamental estimate is then employed in [START_REF] Bénilan | Solutions of the porous medium equation in R N under optimal conditions on initial values[END_REF] for the study of existence theory, in [START_REF] Herrero | The Cauchy problem for u t = ∆u m when 0 < m < 1[END_REF] for L ∞ loc (R n ) estimate for solutions, and in [START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF] for obtaining regularity results for the free boundary of solutions, to name but a few. Later in [START_REF] Lu | Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds[END_REF]

(O, R) × [0, T ] that |∇v| 2 v -β v t v ≤ αβ 2 1 t + C 2 K 2 v 2R,T max + αβ 2 v 2R,T max R 2 C 1 where v 2R,T max = max B(O,2R)×[0,T ] v.
For m ∈ 1 -2 n , 1 , they proved that on B (O, R) × [0, T ], for any γ ∈ (0, 1),

|∇v| 2 v -γ v t v ≥ αγ 2 C 3 1 t + C 4 C 3 K 2 v2R,T max + αγ 2 C 3 v2R,T max R 2 C 5 where v2R,T max = max B(O,2R)×[0,T ] (-v) .
Later in [START_REF] Huang | Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds[END_REF] several results of similar type were obtained by G. Huang, Z. Huang and H. Li. Note that these gradient bounds do not depend on the initial data.

While in [START_REF] Yau | On the Harnack inequalities of partial differential equations[END_REF], S.T. Yau established a similar type of gradient bounds depending on derivatives of initial data for degenerate parabolic equations of the form

∂u ∂t = ∆ (F (u))
with F ∈ C 2 (0, ∞) and F ′ > 0. In particular, as explained in [START_REF] Ma | Gradient estimate for the degenerate parabolic equation u t = ∆F (u) + H(u) on manifolds[END_REF], Yau's result implies that for any function

c (t) ∈ C 1 (0, ∞) satisfying      c (t) ≤ 0 c ′ (t) ≥ 0 |∇v| 2 -2v t + 2m m-1 m v m-2 m-1 ≤ c (t) at t = 0
it holds for all t > 0 that

|∇v| 2 -2v t + 2m m -1 m v m-2 m-1 ≤ c (t) .
Besides gradient estimates of Aronson-Bénilan type, the Hamilton type estimate also plays an important role. It originates from Hamilton [START_REF] Hamilton | A matrix Harnack estimate for the heat equation[END_REF] where it was proved that a positive smooth solution u to the heat equation on a compact manifold without boundary and with Ric≥ -k, k > 0, we have

|∇u| 2 u 2 ≤ 1 t + 2k ln u ∞ u . (5)
This is an upper bound on the gradient of space variables only, hence leading to a different type of Harnack inequalities. As proved by Kotschwar [START_REF] Kotschwar | Hamilton's gradient estimate for the heat kernel on complete manifolds[END_REF], the same result holds for complete noncompact manifolds as well. Later a local version was obtained by Souplet and Zhang in [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF]. As for PME and FDE, L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski [START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF] discussed the case where the initial data is compactly supported. Namely, under the assumption that the initial data u 0 ≥ 0 is integrable and compactly supported, they established that for m > 1, there exists a time

T = T (u 0 ) > 0 and a constant c = c (m, n) > 0 such that |∇v (x, t)| ≤ c v t 1 2 + |x| t
for any t > T and almost every x ∈ R n . Later X. Xu [START_REF] Xu | Gradient estimates for u t = ∆F (u) on manifolds and some Liouvilletype theorems[END_REF] derived a local result on a complete Riemannian manifold with Ric≥ -k for some k ≥ 0. For m > 1, if there exists a constant δ ∈ 0, 4 n-1 such that

1 ≤ v 2R,T max v 2R,T min < 1 1 + δ 4m (n -1) (m -1) + 1 then on B (x 0 , R) × t 0 -T 2 , t 0 |∇v| v 2R,T max (1 + δ) -v ≤ C 6 (m, n)   1 + δ 2ρδR + 1 m-1 m v 2R,T max δρT + k δ   where ρ = 2m - (n -1) (m -1) 2 v 2R,T max (1 + δ) -v 2R,T min v 2R,T min , v 2R,T max = sup B(x0,2R)×[t0-T,t0] v, v 2R,T min = inf B(x0,2R)×[t0-T,t0] v. For m ∈ 1 -4 n+3 , 1 , they obtained that on B (x 0 , R) × t 0 -T 2 , t 0 , |∇v| -v ≤ C 7 (m, n)   1 2R + 1 1-m m v2R,T min T + √ k   where v2R,T min = inf B(x0,2R)×[t0-T,t0] (-v) .
This is a generalization of Li Ma et al. [START_REF] Ma | Gradient estimate for the degenerate parabolic equation u t = ∆F (u) + H(u) on manifolds[END_REF], where the same estimate was derived only for n = 2 or 3 with m ∈ 1 -1 √ n , 1 . In X. Zhu [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds[END_REF], it was proved that for m

∈ 1, 1 + 1 √ 2n+1 , on B (x 0 , R) × t 0 -T 2 , t 0 v 1 4 2-m m-1 |∇v| ≤ C 8 v 2R,T max 1+ 1 4 2-m m-1 1 2R + 1 √ T + √ k with v 2R,T max = sup B(x0,2R)×[t0-T,t0] v.
In X. Zhu [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds[END_REF], a gradient bound for

m ∈ 1 -2 n , 1 was obtained. On B (x 0 , R)× t 0 -T 2 , t 0 |∇v| √ -v ≤ C 9 v2R,T max 1 2R + 1 √ T + √ k where v2R,T max = sup B(x0,2R)×[t0-T,t0] (-v) .
In this paper, let us denote by A R,ǫ x0 the annulus

B (x 0 , R + ǫ) \ B (x 0 , R). Set L = 8 ǫ 2 nǫ 2R + ǫ + 8 (R + ǫ) 2 (2R + ǫ) 2 , h -= m -1 -(n -1) (m -1) 2 , h + = m + 1 -(n -1) (m -1) 2 . ( 6
)
We derived several local and global gradient estimates as follows.

Theorem 1. If u is a positive and bounded solution to (1) on (0,

t 1 )×B (x 0 , R + ǫ) with m ∈ 1 -1 √ n-1 , 1 + 1 √ n-1 , then for any h ∈ (h -, h + ) , ρ ∈ 0, m 2m+h 2h -1 ,
and (T, x) ∈ (0, t 1 ) × B (x 0 , R), we have

u m+h-3 |∇u| 2 (T, x) ≤ I B(x0,R+ǫ) u h 0 ∞ -u h (T, x) hm m -ρ 2m+h 2h T + ρ + 2m+h 2 LM 2 |h| ρ m -ρ 2m+h 2h + |2m -h| l -3 + 6l -1 mM 1 L 2 T 8 (7)
where

l = (h -h -) (h + -h) 3 m -h 2 + 3 , M 1 = sup [0,T ]×A R,ǫ x 0 u 2m+h-2 , M 2 = sup [0,T ]×A R,ǫ x 0 u m+h-1 .
Comparing to the existing work, our result is essentially about estimating the gradient at time t 0 with the information of u during [t 0 -T, t 0 ], while the other results we listed above are using information of u during [t 0 -2T, t 0 ]. Therefore, we have an extra term linear in T . In terms of the constraint on m, one can see that when the dimension n = 1, there is no constraint. When n > 1, comparing to all the existing results we are aware of, our bound is valid for a wider range of m when m < 1. As for the case when m > 1, our range of m is larger than that in [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds[END_REF]. If we look for a bound on the pressure variable, by taking h = 0, the restriction on m becomes identical to the one in [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds[END_REF].

Corollary 2. If u is a positive and bounded solution to (1) on (0, t 1 )×B (x 0 , R + ǫ) with m ∈ 1 -2 n , 1 , and the pressure variable v = u m-1 m-1 , then for any ρ ∈ (0, m), and (T, x) ∈ (0, t 1 ) × B (x 0 , R), we have |∇v| 2 (m -1) v (T, x) ≤ I B(x0,R+ǫ) log u 0 ∞ -log u (T, x) m (m -ρ) T + LM 3 ρ (m -ρ) + m 2 l -3 + 3 2 l -1 mM 1 L 2 T 2 where l = -n (m -1) 2 -2 (m -1) 3 (m + 1) , M 1 = sup [0,T ]×A R,ǫ x 0 u 2m-2 , M 3 = sup [0,T ]×A R,ǫ x 0 ρ log u + (m log u + 1) 2 u m-1
As for the global case, by letting ǫ = R, then R → ∞ and ρ → 0 in [START_REF] Bertsch | Positivity properties of viscosity solutions of a degenerate parabolic equation[END_REF], we obtain a very neat gradient bound.

Corollary 3. If u is a positive and bounded solution to (1) on (0, ∞) × R n and m ∈ 1 -1 √ n-1 , 1 + 1 √ n-1 , then for any h ∈ [h -, h + ] and (T, x) ∈ (0, ∞)×R n , it holds that u m+h-3 |∇u| 2 (T, x) ≤ u h 0 ∞ -u h (T, x) T hm 2 . ( 8
)
First of all, this bound does not depend on the dimension n explicitly. This is because the information about dimension has been incorporated into the initial value and the constraint on h. In fact, [START_REF] Bertsch | Positivity properties of viscosity solutions of a degenerate parabolic equation[END_REF] derived a gradient bound which is independent of the initial data, but dependent on dimension n. Secondly, it naturally shows that when a function touches its maximum, its gradient vanishes. In this sense it is a tight gradient bound. Moreover, when m = 1 and h → 0 + , our results recovers Hamilton's gradient bound for heat equation.

(T, x) ∈ (0, ∞) × R n , it holds that |∇u| 2 u 2 (T, x) ≤ 1 T log u 0 ∞ u (T, x) . ( 9
)
Our last result is an extension of Corollary 3, where the condition that

(h -h -) (h -h + ) < 0, is essential.
We next consider the case where m and h satisfy the condition that

(h -h -) (h -h + ) > 0.
The previous inequality is equivalent to that m ∈ 0, 1 -

1 √ n-1 ∪ 1 + 1 √ n-1 , ∞ and h ∈ R , or m ∈ 1 -1 √ n-1 , 1 + 1 √ n-1 but h ∈ (-∞, h -) ∪ (h + , ∞).
For such m and h, let us define

U ± = mh ± |h| m 2 + (h -h -) (h -h + ) (h -h -) (h -h + ) . ( 10
)
Theorem 5. Let u be a positive and bounded solution to (1) on (0, ∞) × R n . Suppose h ∈ R satisfying the condition that

(h -h -) (h -h + ) > 0,
and

U ≡ u h 0 ∞ u h,T min -1 ∈ (U -, U + ) where u h,T min = inf (t,x)∈[0,T ]×R n u h (t, x) . Then u m+h-3 |∇u| 2 (T, x) ≤ 2 u h 0 ∞ -u h (T, x) mT (h -h -) (h -h + ) U -1 (U -U -) (U + -U ) .
This result, together with Corollary 3 shows that when m ∈ 1 -

1 √ n-1 , 1 + 1 √ n-1
, a gradient bound only depending on the maximum of initial data can be established. While when m is outside of this range, the gradient bound will depend on the minimum of u as well. A similar phenomena has been shown in [START_REF] Xu | Gradient estimates for u t = ∆F (u) on manifolds and some Liouvilletype theorems[END_REF] , but only for m > 1. Nevertheless, the meaning behind this phenomena is not clear to us yet, which is worthy of exploring.

We would like to mention that employing martingale theory to derive gradient estimate is not new, and there is a large number of papers devoted to the study of solutions of PDEs by using stochastic differential equations, for example [START_REF] Picard | Gradient estimates for some diffusion semigroups[END_REF], [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF] and literature therein for a small sample. On the other hand, to the best of authors' knowledge, there are few papers dealing with the kind of nonlinear PDEs by using martingale methods.

The rest of the paper is organized as follows. In Section 2, we illustrate our main idea by establishing the gradient estimate for solution to the heat equation. In Section 3, we prove our local and global estimates.

Gradient estimate for solutions to the heat equation

To illustrate the main idea, let us consider the heat equation in this chapter. Assume u solves

∂u ∂t = ∆u on (0, ∞) × R n u (x, 0) = u 0 (x) on R n . ( 11
)
To avoid technical difficulties, we also assume u is smooth, bounded and has bounded derivatives with respect to space variables up to the second order. It is known that there is a close link between a large class of parabolic PDEs and diffusion processes, in the sense that the differential operator for a PDE can be identified as a generator for a diffusion process. Once this one-to-one correspondence has been established, we are given a way to study a PDE through its diffusion process counterpart, or the other way around. For example, the solution to a PDE can be expressed in terms of the expectation of a diffusion process at a certain time, the so-called path integration, and the transition probability density function of a diffusion process is the kernel of a PDE. For a more comprehensive account of this area, we refer to the book [START_REF] Stroock | Multidimensional diffusion processes[END_REF] by Stroock and Varadhan. Here we also relates our PDE, [START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF], to a diffusion process in the way that we have just explained. For a given point (T, x) ∈ (0, ∞) × R n , define an n-dimensional stochastic process X t by solving the stochastic differential equation

dX α t = √ 2dW α t X α 0 = x α where α ∈ {1, • • • , n} and W = W 1 , • • • , W n is a standard n-dimensional
Brownian motion on the probability space (Ω, F , P). Then we have a progressively measurable function

X from ((0, ∞) × Ω, B ((0, ∞)) ⊗ F ) to R n .
Bearing in mind that the aim is to get an upper bound for |∇u| 2 (T, x), let us consider

the process |∇u| 2 (T -t, X t ) with index t ∈ [0, T ],
that is, the gradient running backward on diffusion process X. The way we composite |∇u| 2 with X is commonly seen. In terms of computation, this will lead to the disappearance of terms containing the time derivative when using Ito formula. Intuitively, this is because on the one hand our diffusion process always starts at the deterministic point that we are interested in and then evolves in a certain random way as t increases, but on the other hand, due to the nature of parabolic PDEs, we need to use the information about the solution before T . So we need the time variable to decrease when t increases. One can observe that |∇u| 2 (Tt, X t ) is a semimartingale. Hence we are encouraged to turn to the theory of martingales, a concept introduced by Paul Lévy in 1930s and greatly developed by J.L Doob in his book [START_REF] Doob | Stochastic processes[END_REF]. First of all, let us decompose the semimartingale |∇u| 2 (Tt, X t ) 0≤t≤T into a sum of a local martingale and a process with finite variation. From now on, when there is no potential confusion, we omit the specification of variables in functions, as we always consider functions running backward on diffusion X. Taking derivatives with respect to x α on both sides of (11) yields

∂ 2 u ∂t∂x α = ∆ ∂u ∂x α . ( 12
)
Since u is smooth by our assumption, we can apply Ito formula on ∂u ∂x α (Tt, X t ) and obtain

d ∂u ∂x α = - ∂ 2 u ∂t∂x α dt + ∆ ∂u ∂x α dt + √ 2 ∂ 2 u ∂x α ∂x β dW β t = √ 2 ∂ 2 u ∂x α ∂x β dW β t
where the last equality results from [START_REF] Chasseigne | Theory of extended solutions for fastdiffusion equations in optimal classes of data. Radiation from singularities[END_REF]. Then, again by Ito formula,

d |∇u| 2 = 2 ∂u ∂x α d ∂u ∂x α + α d ∂u ∂x α = 2 √ 2 ∂u ∂x α ∂ 2 u ∂x α ∂x β dW β t + 2 n α,β=1 ∂ 2 u ∂x α ∂x β 2 dt. ( 13
)
This is the decomposition we are looking for. It can be seen that the finite variation term is non-negative, and the local martingale part is a true martingale under P, a consequence of the boundedness of ∂u ∂x α and

∂ 2 u ∂x α ∂x β , ∀α, β ∈ {1, • • • , n}. This means that |∇u| 2 (T -t, X t ) is a submartingale, which implies that E |∇u| 2 (T -t, X t ) ≥ E |∇u| 2 (T, X 0 ) = |∇u| 2 (T, x)
for any t ∈ [0, T ], where the equality results from the fact that X 0 = x P-almost surely. By integrating both sides on t from 0 to T , we have

ˆT 0 E |∇u| 2 (T -t, X t ) dt ≥ T |∇u| 2 (T, x) . (14)
On the other hand, through a scaling argument, one can see that a bound on |∇u| 2 (T, x) involving T and u ∞ should have the order

O u 2 ∞ T
. Therefore, let us consider the process u 2 (Tt, X t ). By Ito formula,

u 2 (0, X T ) -u 2 (T, X 0 ) = 2 √ 2 ˆT 0 u ∂u ∂x α dW α t + 2 ˆT 0 |∇u| 2 dt. (15)
Since u and ∂u ∂x α , α ∈ {1, • • • , n}, are bounded by our assumption, ´• 0 u ∂u ∂x α dW α t is a true martingale. Hence E ´T 0 u ∂u ∂x α dW α t = 0. It then follows from ( 15) that

E u 2 (0, X T ) -u 2 (T, x) = 2E ˆT 0 |∇u| 2 dt . (16)
By Fubini's theorem, E ´T 0 |∇u| 2 dt = ´T 0 E |∇u| 2 dt. Therefore, ( 14) and ( 16) together yields

|∇u| 2 (T, x) ≤ E u 2 (0, X T ) -u 2 (T, x) 2T .
Since the law of X T is absolutely continuous with respect to Lebesgue measure, u 2 (0, X T ) ≤ u 0 2 ∞ P-almost surely, where • ∞ denotes the essential supremum. Hence we have the following result. Theorem 6. If u solves the Cauchy problem [START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF] and u is smooth, bounded and has bounded derivatives with respect to space variables up to the second order, then for any (T, x) ∈ (0, ∞) × R n , we have

|∇u| 2 (T, x) ≤ u 0 2 ∞ -u 2 (T, x) 2T ≤ u 0 2 ∞ T . (17)
We want to remark here that this method also applies to estimates of gradients of higher orders. To see this, first observe that by considering the function ū (t, x) = u (t + ǫ, x) where ǫ > 0 in the above argument, we get

|∇u| 2 (T, x) ≤ u (ǫ, •) 2 ∞ T -ǫ .
Since ∂u ∂x i also satisfies the heat equation, it holds that

∇ ∂u ∂x i 2 (T, x) ≤ ∂u ∂x i (ǫ, •) 2 ∞ T -ǫ .
Summing over the index i yields

n i,j=1 ∂ 2 u ∂x i ∂x j 2 (T, x) ≤ |∇u| 2 (ǫ, •) ∞ T -ǫ ≤ u 0 2 ∞ ǫ (T -ǫ) ,
where the last inequality results from [START_REF] Hu | BMO Martingales and Positive Solutions of Heat Equations[END_REF]. Then we can minimize the right hand side by choosing ǫ = T 2 , hence obtaining

n i,j=1 ∂ 2 u ∂x i ∂x j 2 (T, x) ≤ 4 u 0 2 ∞ T 2 .
Remark. An approach based on the link between Backward Stochastic Differential Equation (BSDE) and PDE to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds is developed in [START_REF] Hu | BMO Martingales and Positive Solutions of Heat Equations[END_REF].

3 Gradient estimate for positive solutions to PME and FDE

Local gradient estimate

Now we move to the equation [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]. In this section we study the problem on a local scale. Denote by B (x 0 , R + ǫ) a closed ball in R n with center x 0 and radius R + ǫ, where R, ǫ > 0. Let u be a positive and bounded solution to (1) on [0, t 1 ] × B (x 0 , R + ǫ). The positivity of u ensures that no degeneracy of parabolicity would happen in [0, t 1 ] × B (x 0 , R + ǫ). Hence we can use theory about non-degenerate quasilinear parabolic PDE to obtain that u is smooth in [0, t 1 ] × B (x 0 , R + ǫ). Note that at this moment u is defined only on [0, t 1 ] × B (x 0 , R + ǫ). This brings difficulty to our martingale method, as we will consider u running backward on a stochastic process, which takes values on the whole space R n at any time. To get around this obstacle, let ũ be a positive and smooth function with bounded derivatives of all orders defined on [0,

t 1 ] × R n , such that u = ũ on [0, t 1 ] × B (x 0 , R + ǫ) .
Note that as u is strictly positive and smooth on the compact set [0, t 1 ] × B (x 0 , R + ǫ), such ũ exists. It is worthwhile to point out here that the behavior of the extended function ũ outside [0, t 1 ] × B (x 0 , R + ǫ) will not enter into our computation in the sequel, as Li-Yau's localization technique will be adopted.

Next, we take a transformation on ũ by setting

f = ũh h (18)
for some h ∈ R \ {0}. This is a generalization of the transform v = u m-1 m-1 , which repeatedly appears in literature concerning PME and FDE, such as [START_REF] Aronson | Régularité des solutions de l'équation des milieux poreux dans R N[END_REF], [START_REF] Lu | Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds[END_REF], [START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF], [START_REF] Xu | Gradient estimates for u t = ∆F (u) on manifolds and some Liouvilletype theorems[END_REF] and [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds[END_REF]. From (1) one can derive that the so-called pressure variable

v satisfies ∂v ∂t = m (m -1) v∆v + m |∇v| 2 . (19)
We can see that the exponent m in (1) comes down into coefficients in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], and both terms on the right hand side of ( 19) are quadratic, which facilitates many computations. However, this feature is not crucial to our method. Hence we attempt to generalize this transform with [START_REF] Huang | Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds[END_REF]. It turns out that the flexibility in choosing h results in an enlargement of the range of m that our gradient bound is valid for. As we are looking for a gradient bound on a local scale, we are keen to only use the local information about f . For this purpose, we adopt the localization technique of Li and Yau [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF] to introduce a cut-off function φ

∈ C 2 (R n ) satisfying φ (x) = 1 on B (x 0 , R) 0 on B (x 0 , R + ǫ) c , |∆φ| ≤ Lφ 1 2 (20)
and

|∇φ| 2 ≤ Lφ 3 2 , (21)
for some L > 0. Note that such cut-off function exists. One possible choice is

φ (x) =        1 on B (x 0 , R) 0 on B (x 0 , R + ǫ) c R+ǫ R 2 -1 -4 R+ǫ R 2 -x-x0 R 2 4
otherwise and

L = 8 ǫ 2 nǫ 2R + ǫ + 8 (R + ǫ) 2 (2R + ǫ) 2 .
Then function φf , the multiplication of functions φ and f , cuts all the information of f outside [0, t 1 ] × B (x 0 , R + ǫ), while faithfully preserving its behavior in [0, t 1 ] × B (x 0 , R). Therefore, it is this function that we are going to consider in the sequel.

From PDE to SDE

Just as the case of heat equation, let us begin by fixing a point (T, x) ∈ [0, t 1 ] × B (x 0 , R). From (1) and the definition of f , we have on [0,

t 1 ] × R n , ∂φf ∂t = m (hf ) m-1 h φ∆f + m (m -h) (hf ) m-h-1 h φ |∇f | 2 = m (hf ) m-1 h ∆ (φf ) -m (hf ) m-1 h f ∆φ -2m (hf ) m-1 h ∇f • ∇φ + m (m -h) (hf ) m-h-1 h φ |∇f | 2 . ( 22
)
Then let us link PDE [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF] with the diffusion process X = (X t ) 0≤t≤T , whose generator L is given by

L t w (y) = m (hf ) m-1 h (T -t, y) ∆w (y) , ∀w ∈ C 2 0 (R n ) .
L corresponds to the principle part of the differential operator in [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]. Note that by the definition of f , (hf )

m-1 h ≥ 0 since u ≥ 0. By [START_REF] Stroock | Multidimensional diffusion processes[END_REF] the way to obtain X is to solve the stochastic differential equation (SDE)

dX α t = √ 2m (hf ) m-1 2h (T -t, X t ) dW α t X 0 = x for t ∈ [0, T ], where W = W 1 , • • • , W n is a standard n-dimensional
Brownian motion on a probability space (Ω, F , P) and the stochastic integral is in Ito's sense. By our assumption on u, (hf )

m-1 2h
is bounded, smooth and has bounded derivatives. Therefore, by [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] this SDE has a unique strong solution on the time interval [0, T ].

Fundamental decompositions

The next step is to consider various functions running backward on process X. They are f (Tt, X t ), φ (X t ), |∇f | 2 (Tt, X t ) and so on. In fact, we will omit the specification of variables in functions, as all the functions below are compositions on (Tt, X t ). t will always take values in [0, T ], which is exactly the time interval where X lives. One can see that these processes are all semimartingales, waiting for us to decompose and then releasing information.

But unlike the heat equation, [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF] is not linear, a consequence of the nonlinearity of [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]. Therefore, we should be very careful in choosing semimartingales for decomposition. To begin with, let us consider

Y t (φf ) (T -t, X t ) .
By Ito formula,

dY t = √ 2m (hf ) m-1 2h ∂ (f φ) ∂x α dW α t -m (m -h) (hf ) m-h-1 h φ |∇f | 2 dt + ∆φm (hf ) m-1 h f + 2m (hf ) m-1 h ∂f ∂x α ∂φ ∂x α dt. ( 23
)
It is readily seen that

Y T = φf (0, X T ) = φ u h 0 h (X T ) . ( 24
)
Observe that in the finite variation part of ( 23), the term with highest degree

in ∇f is -m (m -h) (hf ) m-h-1 h φ |∇f | 2 (T -t, X t ) dt.
This means that it can be controlled in terms of u 0 ∞ with the help of ( 23). Therefore, it is this term that worth investigation. Define

H t = (hf ) m-h-1 h φ |∇f | 2 (T -t, X t ) , t ∈ [0, T ] .

The function (hf )

m-h-1 h φ |∇f | 2 is smooth on [0, t 1 ] × R n .
Its dynamic only depends on the behavior of u within [0, t 1 ] × B (x 0 , R + ǫ), which is governed by [START_REF] Arnaudon | Li-Yau type gradient estimates and Harnack inequalities by stochastic analysis[END_REF]. Therefore, we can use Ito formula and decompose H into local martingale part and finite variation part as follows.

dH t = d (hf ) m-h-1 h φ α ∂f ∂x α 2 = 2 √ 2m (hf ) 3m-2h-3 2h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α dW β t + √ 2m (m -h -1) (hf ) 3m-4h-3 2h φ |∇f | 2 ∂f ∂x β dW β t + √ 2m (hf ) 3m-2h-3 2h ∂φ ∂x β |∇f | 2 dW β t -m (2m -h + 1) (m -h -1) (hf ) 2m-3h-2 h φ |∇f | 4 dt +2m (m -h -1) (hf ) 2m-2h-2 h ∂φ ∂x α ∂f ∂x α |∇f | 2 dt +m (hf ) 2m-h-2 h ∆φ |∇f | 2 dt -2m (m -1) (hf ) 2m-2-2h h φ∆f |∇f | 2 dt -4m (hf ) 2m-2h-2 h φ ∂f ∂x β ∂ 2 f ∂x β ∂x α ∂f ∂x α dt +4m (hf ) 2m-2-h h ∂φ ∂x β ∂ 2 f ∂x α ∂x β ∂f ∂x α dt +2m (hf ) 2m-h-2 h φ α,β ∂ 2 f ∂x α ∂x β 2 dt. (25)
This is the decomposition under measure P for semimartingale H. But in order to get more flexibility, we need to take advantage of an important tool in stochastic analysis-the change of measure. Let us introduce a family of probability measures on (Ω, F ) depending on a parameter λ ∈ R. Define a measure Q on (Ω, F ) by

dQ dP Ft = exp ˆt 0 √ 2mλ (hf ) m-2h-1 2h ∂f ∂x β dW β -mλ 2 ˆt 0 (hf ) m-2h-1 h |∇f | 2 dt . (26)
Since (hf )

m-2h-1 2h ∂f ∂x β = u m-3 2 ∂u ∂x β , β ∈ {1, • • • , n}
are bounded by our assumption, Novikov's condition [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] is met. So the right hand side of ( 26) is a true martingale under P, which ensures that measure Q is well defined. According to Girsanov's theorem [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], under measure Q, the process

W = W 1 , • • • , W n given by d W β t = dW β t - √ 2mλ (hf ) m-2h-1 2h ∂f ∂x β dt
is an n-dimensional Brownian motion. So from [START_REF] Picard | Gradient estimates for some diffusion semigroups[END_REF], we easily get the decomposition for H under measure Q, which is

dH t = 2 √ 2m (hf ) 3m-2h-3 2h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α d W β t + √ 2m (m -h -1) (hf ) 3m-4h-3 2h φ |∇f | 2 ∂f ∂x β d W β t + √ 2m (hf ) 3m-2h-3 2h ∂φ ∂x β |∇f | 2 d W β t -m (2m -h + 1 -2λ) (m -h -1) (hf ) 2m-3h-2 h φ |∇f | 4 dt +2m (m -h -1 + λ) (hf ) 2m-2h-2 h ∂φ ∂x α ∂f ∂x α |∇f | 2 dt +m (hf ) 2m-h-2 h ∆φ |∇f | 2 dt -2m (m -1) (hf ) 2m-2-2h h φ∆f |∇f | 2 dt +4m (λ -1) (hf ) 2m-2h-2 h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α ∂f ∂x β dt +4m (hf ) 2m-2-h h ∂φ ∂x β ∂ 2 f ∂x α ∂x β ∂f ∂x α dt +2m (hf ) 2m-h-2 h φ α,β ∂ 2 f ∂x α ∂x β 2 dt. (27)
While from [START_REF] Lu | Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds[END_REF], it is easy to see that under measure Q,

dY t = √ 2m (hf ) m-1 2h ∂ (f φ) ∂x α d W α t +m (2λ -m + h) (hf ) m-h-1 h φ |∇f | 2 dt +∆φmh -1 (hf ) m-1+h h dt +2m 1 + λh -1 (hf ) m-1 h ∂f ∂x α ∂φ ∂x α dt. ( 28
)
Now we have ( 28) and ( 27) at hand, which are the fundamental decompositions we are looking for.

Gradient bound for solution u

In contrast to (13) which appears in the case of global estimate for the heat equation, ( 27) is more complicated, thanks to the nonlinearity of PDE ( 1) and the introduction of cut-off function φ. To make use of it, recall that we aim to estimate the first order derivatives of f . Therefore, it is reasonable to get rid off the second order derivatives of f appearing in the decomposition [START_REF] Souplet | Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds[END_REF]. This is done in the following lemma. For simplicity, denote by A R,ǫ x0 the annulus

B (x 0 , R + ǫ) \ B (x 0 , R). Lemma 7. H satisfies dH t ≥ A t d Wt - 1 2 mn (m -1) 2 (hf ) 2m-3h-2 h φ |∇f | 4 dt -m ((2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m)) (hf ) 2m-3h-2 h φ |∇f | 4 dt +2m (2m -h -λ) (hf ) 2m-2h-2 h ∂φ ∂x α ∂f ∂x α |∇f | 2 dt +m (hf ) 2m-h-2 h ∆φ |∇f | 2 dt -2m (hf ) 2m-2-h h φ -1 I A R,ǫ x 0 |∇φ| 2 |∇f | 2 dt ( 29
)
where

A = 2 √ 2m (hf ) 3m-2h-3 2h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α + √ 2m (m -h -1) (hf ) 3m-4h-3 2h φ |∇f | 2 ∂f ∂x β + √ 2m (hf ) 3m-2h-3 2h ∂φ ∂x β |∇f | 2 . (30)
Proof. From ( 27), let us write

dH t A t d Wt + (B + C + D) dt
where

A = 2 √ 2m (hf ) 3m-2h-3 2h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α + √ 2m (m -h -1) (hf ) 3m-4h-3 2h φ |∇f | 2 ∂f ∂x β + √ 2m (hf ) 3m-2h-3 2h ∂φ ∂x β |∇f |
and by Cauchy-Schwartz inequality,

B = α 2m (hf ) 2m-h-2 h φ ∂ 2 f ∂x α ∂x α 2 -2m (m -1) (hf ) 2m-2-2h h φ ∂ 2 f ∂x α ∂x α |∇f | 2 +4m (λ -1) (hf ) 2m-2h-2 h φ ∂ 2 f ∂x α ∂x α ∂f ∂x α ∂f ∂x α +4m (hf ) 2m-2-h h ∂φ ∂x α ∂ 2 f ∂x α ∂x α ∂f ∂x α ≥ -2m (hf ) 2m-2-h h φ -1 I A R,ǫ x 0 α ∂φ ∂x α 2 ∂f ∂x α 2 -2m (λ -1) 2 (hf ) 2m-3h-2 h φ α ∂f ∂x α 4 -4m (λ -1) (hf ) 2m-2h-2 h α ∂φ ∂x α ∂f ∂x α 3 +2m (m -1) (hf ) 2m-2-2h h |∇f | 2 ∂φ ∂x α ∂f ∂x α + λ -1 - 1 4 n (m -1) 2m (m -1) (hf ) 2m-2-3h h φ |∇f | 4 , C = α =β 2m (hf ) 2m-h-2 h φ ∂ 2 f ∂x α ∂x β 2 +4m (λ -1) (hf ) 2m-2h-2 h φ ∂ 2 f ∂x α ∂x β ∂f ∂x α ∂f ∂x β +4m (hf ) 2m-2-h h ∂φ ∂x β ∂ 2 f ∂x α ∂x β ∂f ∂x α ≥ α =β -2m (λ -1) 2 (hf ) 2m-3h-2 h φ ∂f ∂x α ∂f ∂x β 2 -2m (hf ) 2m-2-h h φ -1 I A R,ǫ x 0 ∂φ ∂x β ∂f ∂x α 2 -4m (λ -1) (hf ) 2m-2h-2 h ∂φ ∂x β ∂f ∂x α ∂f ∂x α ∂f ∂x β and D = -m (2m -h + 1 -2λ) (m -h -1) (hf ) 2m-3h-2 h φ |∇f | 4 +m (hf ) 2m-h-2 h ∆φ |∇f | 2 +2m (m -h -1 + λ) (hf ) 2m-2h-2 h ∂φ ∂x α ∂f ∂x α |∇f | 2 .
Then adding these three inequalities gives us (29).

To deal with the finite variation part in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear parabolic equations of porous medium type[END_REF], let us assume that

1 2 n (m -1) 2 + (2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m) < 0. (31)
This allows us to get the following estimate.

Lemma 8. Under assumption (31), we have

E Q ˆT 0 H s ds ≥ H 0 T - 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 T 2 2 (32)
where

l = 1 2 n (m -1) 2 + (2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m) -3 2 |2m -h -λ| -3 2 (33)
and

M 1 = sup [0,T ]×A R,ǫ x 0 u 2m+h-2 (t, y) .
Proof. By Cauchy-Schwartz inequality, for any positive l 1 , l 2 , l 3 and l 4 ,

(hf ) 2m-2h-2 h ∂φ ∂x α ∂f ∂x α |∇f | 2 ≤ 1 2 l 1 (hf ) 2m-3h-2 h φ |∇f | 4 + 1 2 l -1 1 (hf ) 2m-h-2 h φ -1 |∇φ| 2 |∇f | 2 ≤ 1 2 l 1 (hf ) 2m-3h-2 h φ |∇f | 4 + 1 4 l -1 1 l 2 (hf ) 2m-3h-2 h φ |∇f | 4 + 1 4 l -1 1 l -1 2 (hf ) 2m+h-2 h φ -3 |∇φ| 4 , (hf )
2m-h-2 h φ -1 |∇φ| 2 |∇f | 2 ≤ 1 2 l 3 (hf ) 2m-3h-2 h φ |∇f | 4 + 1 2 l -1 3 (hf ) 2m+h-2 h φ -3 |∇φ| 4 and (hf ) 2m-h-2 h ∆φ |∇f | 2 ≤ 1 2 l 4 (hf ) 2m-3h-2 h φ |∇f | 4 + 1 2 l -1 4 (hf ) 2m+h-2 h φ -1 |∆φ| 2 .
Plugging them into (29) in the above lemma yields

dH t ≥ A t d Wt - 1 2 mn (m -1) 2 (hf ) 2m-3h-2 h φ |∇f | 4 dt -m ((2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m)) (hf ) 2m-3h-2 h φ |∇f | 4 dt -2m |2m -h -λ| 1 2 l 1 + 1 4 l -1 1 l 2 (hf ) 2m-3h-2 h φ |∇f | 4 dt -ml 3 (hf ) 2m-3h-2 h φ |∇f | 4 dt - 1 2 ml 4 (hf ) 2m-3h-2 h φ |∇f | 4 dt - 1 2 ml -1 4 (hf ) 2m+h-2 h φ -1 |∆φ| 2 dt - 1 2 m |2m -h -λ| l -1 1 l -1 2 (hf ) 2m+h-2 h φ -3 |∇φ| 4 dt -ml -1 3 (hf ) 2m+h-2 h φ -3 |∇φ| 4 dt.
For simplicity, set

l 2 1 = l 2 = l 2 3 = l 2 4 = l 2 . Then we have dH t ≥ A t d Wt - 1 2 mn (m -1) 2 (hf ) 2m-3h-2 h φ |∇f | 4 dt -m ((2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m)) (hf ) 2m-3h-2 h φ |∇f | 4 dt - 3 2 m |2m -h -λ| l (hf ) 2m-3h-2 h φ |∇f | 4 dt - 3 2 ml (hf ) 2m-3h-2 h φ |∇f | 4 dt - 1 2 m |2m -h -λ| l -3 + 2l -1 (hf ) 2m+h-2 h φ -3 |∇φ| 4 dt - 1 2 ml -1 (hf ) 2m+h-2 h φ -1 |∆φ| 2 dt.
By estimates for the cut-off function φ in ( 20) and ( 21),

dH t ≥ A t d Wt - 1 2 mn (m -1) 2 (hf ) 2m-3h-2 h φ |∇f | 4 dt -m ((2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m)) (hf ) 2m-3h-2 h φ |∇f | 4 dt - 3 2 m |2m -h -λ| l (hf ) 2m-3h-2 h φ |∇f | 4 dt - 3 2 ml (hf ) 2m-3h-2 h φ |∇f | 4 dt - 1 2 m |2m -h -λ| l -3 + 3l -1 (hf ) 2m+h-2 h I A R,ǫ x 0 L 2 dt.
Now, by assumption [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type[END_REF], we can choose a positive l small enough such that

- 1 2 mn (m -1) 2 -m ((2m -h + 1 -2λ) (m -h -1) + 2 (λ -1) (λ -m)) = 3 2 |2m -h -λ| + 3 2 ml.
Hence we have for any t ∈ [0, T ],

dH t ≥ A t d Wt - 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 dt
where

M 1 = sup [0, T ] × A R,ǫ x0 (hf ) 2m+h-2 h (t, y) = sup [0,T ]×A R,ǫ
x 0 u 2m+h-2 (t, y) .

This gives

H s -H 0 ≥ ˆs 0 A t d Wt - 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 s (34)
for any s ∈ [0, T ]. According to our assumption on u, A, defined by [START_REF] Vázquez | Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations[END_REF], is bounded. So ´• 0 A s d Ws is a true martingale under Q. Therefore we can take expectation on both sides of [START_REF] Yau | On the Harnack inequalities of partial differential equations[END_REF] and obtain

E Q [H s ] ≥ H 0 - 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 s. (35)
Integrating both sides with respect to s on [0, T ] yields [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF].

On the other hand, by using [START_REF] Lu | Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds[END_REF], we have an upper bound on the left hand side of [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF].

Lemma 9. Assume 2λ -m + h > 0. ( 36
)
Then for any ρ > 0 such that

2λ -m + h -ρ 1 + λh -1 > 0, (37)
we have

E Q ˆT 0 H t dt ≤ E Q [Y T ] -Y 0 + h -1 + ρ -1 1 + λh -1 mLM 2 T m (2λ -m + h -ρ |1 + λh -1 |) (38) with M 2 = sup [0,T ]×A R,ǫ
x 0 u m+h-1 (t, y) .

Proof. Recall [START_REF] Stroock | Multidimensional diffusion processes[END_REF], which is the fundamental decomposition of Y under measure Q. Then by property (20) of cut-off function φ and Cauchy-Schwartz inequality, for any ρ > 0,

dY t ≥ √ 2m (hf ) m-1 2h ∂ (f φ) ∂x α d W α t +m (2λ -m + h) (hf ) m-h-1 h φ |∇f | 2 dt -Lφ 1 2 I A R,ǫ x 0 m h -1 (hf ) m-1+h h dt -ρm 1 + λh -1 (hf ) m-1-h h φ |∇f | 2 dt -ρ -1 m 1 + λh -1 (hf ) m-1+h h I A R,ǫ x 0 φ -1 |∇φ| 2 dt ≥ √ 2m (hf ) m-1 2h ∂ (f φ) ∂x α d W α t +m 2λ -m + h -ρ 1 + λh -1 H t dt -h -1 + ρ -1 1 + λh -1 mLM 2 dt (39)
where

M 2 = sup [0,T ]×A R,ǫ x 0 (hf ) m+h-1 h (t, y) = sup [0,T ]×A R,ǫ
x 0 u m+h-1 (t, y) .

According to our assumption on u, (hf )

m-1 2h ∂(f φ) ∂x α , α ∈ {1, • • • , n} are all bounded, which ensures that ´• 0 (hf ) m-1 2h ∂(f φ) ∂x α d W α t is a Q martingale. Hence E Q [Y T -t1 ] -Y 0 ≥ m 2λ -m + h -ρ 1 + λh -1 E Q ˆT -t1 0 H t dt -h -1 + ρ -1 1 + λh -1 mLM 2 (T -t 1 ) .
Dividing both sides by m 2λm + hρ 1 + λh -1 yields (38). Now, based on ( 32) and (38), we are ready to obtain a gradient bound for u.

Proposition 10. Let the assumptions in Lemma 8 and Lemma 9 be satisfied, then for any (T, x) ∈ (0, t 1 ) × B (x 0 , R),

u m+h-3 |∇u| 2 (T, x) ≤ E Q φu h 0 (X T ) -u h (T, x) hm (2λ -m + h -ρ |1 + λh -1 |) T + h -1 + ρ -1 1 + λh -1 LM 2 2λ -m + h -ρ |1 + λh -1 | + 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 T 2 . ( 40
)
Proof. Combining ( 32) with (38), we have

H 0 ≤ E Q [Y T ] -Y 0 + h -1 + ρ -1 1 + λh -1 mLM 2 T m (2λ -m + h -ρ |1 + λh -1 |) T + 1 2 |2m -h -λ| l -3 + 3 2 l -1 mM 1 L 2 T 2 .
Then we arrive at (40) by noting that

H 0 = (hf ) m-h-1 h φ |∇f | 2 (T, x) , Y 0 = φf (T, x) = φ u h h (T, x)
and

E Q [Y T ] = E Q φ u h 0 h (X T ) .
Let us revise the assumptions we have made, that is, [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type[END_REF] and [START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds[END_REF]. They impose a restriction on the choice of m. Let us look at [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type[END_REF] first. It is equivalent to

2λ 2 -2 (2m -h) λ + 1 2 n (m -1) 2 + (2m -h + 1) (m -h -1) + 2m < 0 (41)
To let the left hand side attain the minimum, we take λ = 2m-h 2 . Then (41) is reduced to

h 2 -2mh + n (m -1) 2 + 2 (m -1) < 0. (42)
Notice that after setting λ = 2m-h 2 , condition ( 36) is automatically satisfied. Moreover, (42) holds if and only if m ∈ 1 -

1 √ n-1 , 1 + 1 √ n-1 , and h ∈ m -1 -(n -1) (m -1) 2 , m + 1 -(n -1) (m -1) 2 .
Then we arrive at Theorem 1 by substituting λ = 2m-h 2 into (37) and [START_REF] Xu | Gradient estimates for u t = ∆F (u) on manifolds and some Liouvilletype theorems[END_REF], and noting that E Q φu h 0 (X T ) ≤ I B(x0,R+ǫ) u h 0 ∞ .

Gradient bound for pressure variable

v = u m-1 m-1
Since most of the existing results are in terms of |∇v| 2 v , we are tempted to work out a bound on it with our method. Notice that |∇v| 2 v = (m -1) u m-3 |∇u| 2 , which corresponds to the LHS of [START_REF] Bertsch | Positivity properties of viscosity solutions of a degenerate parabolic equation[END_REF] in Theorem 1 when h = 0. However, the RHS of (7) will explode when h → 0. Therefore, we can not use the result in Theorem 1 directly.

Let us modify the estimate we obtained from the decomposition of Y , so that the bound does not explode when h → 0. The idea is to decompose

Y t -φ h (X t ) instead of Y t . This is because Y t - φ h (X t ) = φ u h -1 h → φ log u, which is finite. Lemma 11. Assume 2λ -m + h > 0. (43)
Then for any ρ > 0 such that

2λ -m + h -ρ > 0, (44)
we have

E Q ˆT 0 H t dt ≤ E Q Y T -φ h (X T ) -Y 0 + φ h (x) + Lmρ -1 M 3 T m (2λ -m + h -ρ) (45)
where

M 3 = sup [0,T ]×A R,ǫ x 0 ρ u h -1 h u h + λ u h -1 h + u h 2 u m-h-1 (t, y) .
Proof. From [START_REF] Stroock | Multidimensional diffusion processes[END_REF],

d Y t - φ h (X t ) = √ 2m (hf ) m-1 2h φ ∂f ∂x α + f - 1 h ∂φ ∂x α d W α t +m (2λ -m + h) (hf ) m-h-1 h φ |∇f | 2 dt +∆φm f - 1 h (hf ) m-1 h dt +2m λ f - 1 h + hf (hf ) m-h-1 h ∂f ∂x α ∂φ ∂x α dt. (46)
Then by Cauchy-Schwartz inequality, for any ρ > 0,

d Y t - φ h (X t ) ≥ √ 2m (hf ) m-1 2h φ ∂f ∂x α + f - 1 h ∂φ ∂x α d W α t +m (2λ -m + h) (hf ) m-h-1 h φ |∇f | 2 dt -Lφ 1 2 I A R,ǫ x 0 m f - 1 h (hf ) m-1 h dt -ρm (hf ) m-h-1 h φ |∇f | 2 dt -ρ -1 m λ f - 1 h + hf 2 (hf ) m-h-1 h I A R,ǫ x 0 φ -1 |∇φ| 2 dt ≥ √ 2m (hf ) m-1 2h φ ∂f ∂x α + f - 1 h ∂φ ∂x α d W α t +m (2λ -m + h -ρ) H t dt -Lmρ -1 ρ f - 1 h hf + λ f - 1 h + hf 2 (hf ) m-h-1 h dt. Let M 3 = sup [0,T ]×A R,ǫ x 0 ρ u h -1 h u h + λ u h -1 h + u h 2 u m-h-1 (t, y) .
According to our assumption on u, (hf )

m-1 2h φ ∂f ∂x α + f -1 h ∂φ ∂x α , α ∈ {1, • • • , n} are all bounded, which ensures that the local martingale part of Y t -φ h (X t ) is a true Q martingale. Hence E Q Y T - φ h (X T ) -Y 0 + φ h (x) ≥ m (2λ -m + h -ρ) E Q ˆT 0 H t dt -Lmρ -1 M 3 T,
which completes the proof.

By combining (45) with [START_REF] Vázquez | The porous medium equation. Mathematical theory[END_REF] in the way as in the proof of Theorem 1 and also taking λ = 2m-h 2 , we are able to get the following gradient bound. Theorem 12. If u is a positive and bounded solution to (1) on (0, t 1 )×B (

x 0 , R + ǫ) with m ∈ 1 -1 √ n-1 , 1 + 1 √ n-1 , then for any h ∈ m -1 -(n -1) (m -1) 2 , m + 1 -(n -1) (m -1) 2 ,
ρ ∈ (0, m), and (T, x) ∈ (0, t 1 ) × B (x 0 , R), we have

u m+h-3 |∇u| 2 (T, x) ≤ E Q φ u h 0 -1 (X T ) -u h (T, x) -1 hm (m -ρ) T + LM 3 ρ (m -ρ) + 1 4 |2m -h| l -3 + 3 2 l -1 mM 1 L 2 T 2 where l = -1 2 n (m -1) 2 + (h -1) (m -1) -1 2 h 2 + h 3 4 |2m -h| + 3 2 , M 1 = sup [0,T ]×A R,ǫ x 0 u 2m+h-2 , M 3 = sup [0,T ]×A R,ǫ x 0 ρ u h -1 h u h + 2m -h 2 u h -1 h + u h 2 u m-h-1 (t, y) ,
and L is a constant depending on the cut-off function φ by ( 20) and [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type. Izdat[END_REF].

By letting h → 0, we immediately get Corollary 2.

Global gradient estimate

In this section, we consider a positive and bounded solution u to (1) on (0, ∞) × R n . First of all, as a direct consequence of the main result in the last section, we can get a global gradient bound from the local bound by letting the radius of the local ball tend to infinity.

From local bound to global bound

Since u is positive and bounded on (0, ∞) × R n , the local result in Theorem 1 holds for any t 1 , R and ǫ. By substituting ǫ = R into (7), and then taking R → ∞ on both sides, we obtain

u m+h-3 |∇u| 2 (T, x) ≤ E Q u h 0 (X T ) -u h (T, x) T hm m -ρ 2m+h 2h
where we have used the fact that when ǫ

= R lim R→∞ L = lim R→∞ 8 R 2 nR 2R + R + 8 (2R) 2 (3R) 2 = 0 and lim R→∞ φ = 1.
By letting ρ = 0, we have

u m+h-3 |∇u| 2 (T, x) ≤ E Q u h 0 (X T ) -u h (T, x) T hm 2 . ( 47
)
Note that in the local case, due to the existence of l -1 in (7), h is not allowed to touch the boundary of the open interval

m -1 -(n -1) (m -1) 2 , m + 1 -(n -1) (m -1) 2 .
But now we can take limit of h to

m -1 -(n -1) (m -1) 2 or m + 1 -(n -1) (m -1) 2
on both sides of (47). Since

E Q u h 0 (X T ) = E Q u h 0 (X T ) I {u0(XT )≥1} + E Q u h 0 (X T ) I {u0(XT )<1} ,
we can apply monotone convergence theorem and bounded convergence theorem on these two terms respectively. Therefore, (47) holds also for h = m -1 -(n -1) (m -1) 2 and h = m + 1 -(n -1) (m -1) 2 .

Hence we obtained Corollary 3. Note that when m = 1 in Corollary 3, h is allowed to approach to 0 + . Since u 0 > 0, by Jensen's inequality, it holds that

lim h→0 + E Q u h 0 (X T ) -u h (T, x) T h = lim h→0 + E Q u h 0 (X T ) -1 T h -lim h→0 u h (T, x) -1 T h = E Q [log u 0 (X T )] -log u (T, x) T = E Q log u0(XT ) u(T,x) T ≤ 1 T log E Q [u 0 (X T )] u (T, x) ≤ 1 T log u 0 ∞ u (T, x)
Therefore, by letting h → 0 in (8), we have Corollary 4.

Negative finite variation part

Now we proceed from fundamental decompositions in a different way in order to get a global gradient bound valid for a wider class of m. From ( 29) and ( 28), we have the global version of two fundamental decompositions. Then (49) becomes

dH t ≥ A t d Wt
dH t ≥ A t d Wt -L 1 (hf ) 2m-3h-2 h |∇f | 4 dt = A t d Wt -L 1 h -1 Y -1 t H 2 t dt. (50)
Previously, in order to obtain the submartingale property of H, we always assume L 1 ≤ 0, resulting in a constraint on m. In this section, we consider the situation when L 1 > 0.

Instead of deriving a gradient bound directly, we seek for an integral inequality satisfied by the gradient bound as a function of t. The arguments are based on the following estimate.

Lemma 13. H • e ´• 0 L1h -1 Y -1 r
Hr dr is a submartingale. Moreover, Then

m (2λ -m + h) E Q ˆT s H t F s = E Q [ Y T | F s ] -Y s . (51)
g (0) ≤ E Q YT Y0 -1 m (2λ -m + h) G (0) , (53)
G (T ) = 0. Moreover, for any s ∈ (0, T )

G ′ (s) = -1 + L 1 h -1 g (s)
ˆT s e -´t s L1h -1 g(u)du dt

≤ -1 + L 1 h -1 E Q YT Ys F s -1 ∞ m (2λ -m + h) .
As a consequence,

G (0) = G (T ) - ˆT 0 G (s) ′ ds ≥ G (T ) + ˆT 0 m (2λ -m + h) -L 1 h -1 E Q YT Ys F s -1 ∞ m (2λ -m + h) ds = 1 m (2λ -m + h) ˆT 0 m (2λ -m + h) -L 1 h -1 E Q Y T Y s F s -1 ∞ ds.
This, together with (53) yields

g (0) ≤ E Q YT Y0 -1 ´T 0 m (2λ -m + h) -L 1 h -1 E Q YT Ys F s -1 ∞ ds
as long as

ˆT 0 m (2λ -m + h) -L 1 h -1 E Q Y T Y s F s -1 ∞ ds > 0.
This means

u m+h-3 |∇u| 2 (T, x) ≤ E Q u h 0 (X T ) -u h (T, x) ´T 0 mh (2λ -m + h) -L 1 E Q u h 0 (XT ) u h (T -s,Xs) F s -1 ∞ ds .
Define u h,T min = inf (t,x)∈[0,T ]×R n u h (t, x) . It then follows that when

mh (2λ -m + h) -L 1 u h 0 ∞ u h,T min -1 > 0, (54)
we have

u m+h-3 |∇u| 2 (T, x) ≤ u h 0 ∞ -u h (T, x) T mh (2λ -m + h) -L 1 u h 0 ∞ u h,T min -1 . (55)
To maximize mh (2λm + h) -L 1 . Then (55) becomes

u m+h-3 |∇u| 2 (T, x) ≤ u h 0 ∞ -u h (T, x) mT mh + mh -m + 1 -h 2 2 -1 2 n (m -1) 2 u h 0 ∞ u h,T min -1 + h 2 2 u h 0 ∞ u h,T min -1 -1 ,
and the constraint (54) becomes

mh+ mh -m + 1 - h 2 2 - 1 2 n (m -1) 2 u h 0 ∞ u h,T min -1 + h 2 2 u h 0 ∞ u h,T min -1 -1 > 0.
By rewriting them in terms of U , U ± and h ± , we have Theorem 5.

  , a local version of Aronson-Bénilan estimate was obtained by P. Lu, L. Ni, J. L. Vazquez and C. Villani. They studied the same problem posed on a local ball of a Riemannian manifold. Let B (O, 2R) denote a ball with center O and radius 2R > 0 . Assume that u is a positive solution to (1) on B (O, 2R) × [0, T ] and the Ricci curvature Ric≥ -(n -1) K 2 on B (O, 2R) for some K ≥ 0. They showed that for any m > 1 and β > 1, it holds on B

Corollary 4 .

 4 (Hamilton's gradient estimate) If u is a solution to the heat equation on (0, ∞) × R n with positive and bounded initial data, then for any

L 1 =

 1 (2mh + 1 -2λ) (mh -1) + 2 (λ -1) (λm)) (hf ) ∂x α d W α t + m (2λm + h) (hf ) m ((2mh + 1 -2λ) (mh -1) + 2 (λ -1) (λm)) + 1 2 mn (m -1) 2 ,

  Q e ´t 0 L1h -1 Y -1 r Hrdr H t F s ≥ e ´s 0 L1h -1 Y -1 r Hr dr H s , which implies E Q e ´t s L1h -1 Y -1 r Hr dr H t F s ≥ H s . Hence E Q [ H t | F s ] ≥ e -´t s L1h -1 g(r)dr H s . ˆT s H t dt F s ≥ H s ˆT s e -´t s L1h -1 g(r)dr dt.Together with (51), it follows thatm (2λm + h) H s ˆT s e -´t s L1h -1 g(r)dr dt ≤ E Q [ Y T | F s ] -Y s ´T s e -´t s L1h -1 g(u)du dt .Let us solve this integral inequality. SetG (s) =ˆT s e -´t s L1h -1 g(u)du dt.

	Then we have
	E Q
	which leads to
	g (s) ≤	E Q YT Ys F s -1

Now let us define

g (t) = Y -1 t H t ∞ .

Lemma 14. Let u be a positive and bounded solution to (1) on (0, ∞) × R n .

Assume L 1 ≥ 0 and 2λm + h > 0. Then

g (s) ≤ E Q YT Ys F s -1 ∞ m (2λm + h) ´T s e -´t s L1h -1 g(u)du dt . (

52

) Proof. Since e ´t 0 L1h -1 Y -1 r Hr dr H t t∈[0,T ] is a submartingale by Lemma 13, for any 0 ≤ s < t ≤ T E ∞ m (2λm + h)


Introduction Shoreline lenght VS area

Geographical and historical settings

In France, majority (98-99%) of SWB are man-made. In Estonia, 56% of total inventory of SWB are man-made and 44% natural SWB.

Only ~ 1500 of natural water bodies are lakes (mean surface area -30 ha excluding L. Peipsi), remaining SWB are small bog pools (mean surface area -0.03 ha). • In Estonia the density of the waterbodies is 2.3 per km2 and in France 1.1 per km2. The median size of standing water bodies (SWB) in Estonia and France is smaller (0.026 ha and 0.071 ha, respectively) than the detection limit (0.2 ha) of the most recent global inventory based on GLOWABO data from satellite imagery (Verpoorter & al., 2014). Our analysis of the abundance and distribution EU water bodies suggests that these global counts underrepresent the hydrologically complex terrain of European territory.

• Our present estimates of all EU inventory of water bodies in all size ranges based on OSM database is ~1 million. The comparison of ground-validated data from national inventories and OSM database shows that reliability of OSM data varies between countries. The number of small standing water bodies remains underestimated, especially in size class below 0.1 ha (the average representation of SWB for EU level in size range 0.01 -0.1 ha is 18%, below 0.01 ha is inaccurate).

• The calculated perimeter in Estonian and French SWB is consistent with a hypothesis by Verpoorter et al. (2014) that the most of the global land-lake interface is in small SWB.

• The abundance and size distribution of SWB in Estonia and France shows the relative importance of very small and man-made water bodies in EU context. The abundance of man-made SWB is globally less studied, especially in the smallest size ranges, but they have important role in sustaining aquatic life while at the same time supporting human activities. Because of their small size and catchments, small SWB are far easier and cheaper to protect and restore than larger SWB.

• All SWB, including small and man-made ponds, play an important role in ecosystem services and require careful management to avoid hydrological and environmental deterioration.

• Combining the national inventories of SWB proves to be the best available solution for estimating the regional inventories of SWB. For each geographical region specific function must be used in order to calculate the total number of SWB.

• The majority of terrestrial standing water bodies (SWB) are small in size, however, their abundance and distribution is not fully known and they are under-represented in legislation. For example, the EU Water Framework Directive 2000/60/EC proposes a threshold surface area for reporting as 50 ha.

• Small SWB can be well represented in the state-level inventories and studies, e.g. <10 ha in Meybeck, 1995, <1 ha in Rjanžin 2005, or <0.05 ha in Kuusisto & Raatikainen 1988, but these studies are region-specific and not suitable for estimating global inventories. Also, there is a bias towards studies of natural lakes while abundance of man-made water bodies is understudied.

• The models for global inventories of SWB are so far not sufficiently designed for estimating the relative abundance of small SWB (below 0.2 ha) and provide differing estimates (Downing et al. 2006;Verpoorter et al. 2014).

• In this pilot study, we :

(1) suggest a bottom-up approach for estimating the number of SWB at EU-level that combines the ground-validated data on water bodies from state inventories and data from peer-generated map databases (Bartout & Touchart, 2013).

(2) assess the inventories and relative importance of small terrestrial water bodies of two different countries, Estonia and France.

Repeated random check of OSM and Estonian national inventory datasets against aerial imagery reveals that both dataset have some limitations. In both cases, some SWB remain unrepresented (majority of bog pools is missing from OSM).

Conclusions

Quality of data

The majority of lake basins formed during the ice retreat after the Last Glacial Maximum. Besides forests (50%), wetlands are the most common landscape feature covering 20% of the territory. Peat-rich mires with characteristic bog pools are abundant. Creation of man-made lakes and ponds has been relatively common throughout the centuries. The man-made water bodies serve range of ecological, cultural and agricultural purposes.

Except few mountain margins, France was not glaciated during the Würm period. Forests account for 28% of land area. The territory has extensive river systems (Seine, Loire, Garonne and Rhone). Most of the water bodies are artificial (~98%), smaller than Nordic lakes and built on fluvial continuum. The tradition of creating the fishponds was developed in Middle Ages and today they provide variety of agricultural functions. Historically, France has also a tradition of forming artificial wetlands.

The comparison of size classes of SWB in Estonian Topographic Database (ETD) and French national inventory. The green color indicates the calculated total shoreline lenght (perimeter) and blue color indicates the total area.

According to the analysis of the OpenStreetmap (OSM) database (Partout, accepted) the number os SWB in EU is 1 041 739. This number covers in average over 80% of SWB in the size class >10 ha, but it is strongly underestimated in smaller size OpenStreetMap (OSM) is a map database built by a community of mappers that contribute and maintain data.

Abundance and size distribution

  Table shows the total inventory of SWB in Estonia divided in size ranges.

VS man-made SW B at EU scale

			<0.01 ha	0.01-0.1 ha	0.1-1 ha				1-10 ha	10-100 ha	>100 ha	12.5%	1.9% 0.3%	The distribution of the
		Estonia								69.3%	15.9%	0.05%	SWB in Estonia and France. Maps are showing SWB
										abundance in different
			16 344	71 273	12 831				1 999	348	52	size ranges. Every dot
										on the map represents
		France								34.5%	50.6%	7.2% 7.0%	0.6% 0.05%	single SWB. The number in corners indicates total counts. Pie-plots are summarising the
										findings.
			41 876	302 994	206 412				43 068	3 727	294	<0.01 ha 0.01-0.1 ha 0.1-1 ha 1-10 ha 10-100 ha >100 ha
	Estonia (45 215	EU territory of
	2 km ) belongs to	France (547 030
	the Boreal region	2 km ) possesses a
	of Europe.	wide variety of
	The surface	coastal, lowland
	topography is flat	and mountain
	( 60% of the	landscapes.
	territory is < 50 m	Estonia		France
	a.s.l).	<0.01 ha	110 ha (0.16%) 532 km (3.57%)	204 ha (0.04%) 1215 km (0.90%)
		0.01-0.1 ha	2 359 ha (3.33%) 5 518 km (36.99%)	12 774 ha (2.81%) 25 268 km (18.63%)
		0.1-1 ha	3 284 ha (4.63%) 3 659 km (24.52%)	66 594 ha (14.64%) 53 217 km (39.23%)
		1-10 ha	5 946 ha (8.39%) 2 191 km (14.69%)	112 404 ha (24.70%) 36 291 km (26.76%)
		10-100 ha	9 127 ha (12.88%) 1 273 km (8.53%)	86 924 ha (19.10%) 12 305 km (9.07%)		Perimeter, km	Area, ha	Count
					<0.01 ha	291	241	60	51 8 712 7 632
					0.01-0.1 ha 2 443 3 076	973 1 386 31 791 39 482
		>100 ha*	932 km (11.70%) 50 049 ha (70.62%)	17 6103 ha (38.70%) 7 343 km (5.41%)	0.1-1 ha 1-10 ha 10-100 ha	1 053 2 607 976 1 213 3 085 2 857 866 2 421 3 450 9 381 963 1 036 766 500 6 268 2 759 223 125
					>100 ha*	626	305 38 032 12 017	45	7
					Sum	6 156 7 942 49 284 21 490 45 184 57 663
					Min.	0.02	0.01	0.00	0.00	-	-
					Max.	130	211 26 901 10 553	-	-
					Median	0.06	0.07	0.02	0.03	-	-
					Mean	0.14	0.14	1.09	0.37	-	-
									* Excluding L. Peipsi	Estonian Topographic Database
										OpenStreetMap	Missing from both

* excluding L. Peipsi (area 354 105 ha; perimeter 814 km) Natural Download poster as pdf. Nat. M.-m. Nat. M.-m. Nat. M.-m.

Acknow ledgments

European perspectives on regional estimates of standing water bodies and the relevance of man-made ponds Institute of Ecology at Tallinn University, Estonia

Laboratoire de géographie CEDETE de l'Université d'Orléans, France Contact: jaanus.terasmaa@tlu.ee; pascal.bartout@univ-orleans.fr


François Ravetta

Jacques Pelon

Jérôme Bureau

Vincent Mariage

Lionel Doppler

André Hollstein

J Fischer

Analysis of atmospheric properties and surface radiation budget using radiative transfer code MOMO

Extension of radiative transfer code MOMO and Validation

Analysis of atmospheric properties and surface radiation budget using radiative transfer code MOMO 1-D radiative transfer code MOMO (Matrix-Operator Model, [R1]), has been extended from [0.2 -3.65 μm] band to the whole [0.2 -100 μm] spectrum [R2]. MOMO can now be used for computation of full range radiation budgets (shortwave and longwave). This extension to the longwave part of the electromagnetic radiation required to consider radiative transfer processes that are features of the thermal infrared: the spectroscopy of the water vapor self-continuum absorption at 12 μm and the emission of radiation by gases, aerosol, clouds and surface. MOMO's spectroscopy module, CGASA (Coefficient of Gas Absorption), has been developed for computation of gas extinction coefficients, considering continua and spectral line absorption. The extension of the code allows the utilization of MOMO as forward model for remote sensing algorithms in the full range spectrum. Another application is full range radiation budget computations (heating rates or forcings).

F. Ravetta, J. Pelon, J. Bureau, V. Mariage,L. Doppler, A. Hollstein, J. Fischer

Application to IAOOS project combining ground-based and space observations

Conclusions and Perspectives

Cloud and aerosols in the arctic atmosphere are key parameters in the surface energy budget but their properties and their modifications are poorly known. The Artic region is experiencing a strong warming and few measurements are currently performed in this region to develop a proper knowledge of the couplings between ocean and atmosphere. Jointky led by LOCEAN and LATMOS, the aim of the IAOOS project (Ice -Atmosphere -Arctic Ocean Observing System) is to bridge this gap by implementing multi-disciplinary autonomous systems on a network of buoys in central arctic.

The atmospheric measurements within IAOOS are aiming at bringing new information on cloud and aerosol properties in the low atmosphere setting a network of small automated profiling lidar systems embarked on buoys.

It is complemented by in situ and radiometric observations which contribute to satellite ground truth.

Example of analyses started in this frame to link satellite and surface observations are discussed below. Two main approaches are considered :

-retrieval of cloud properties from RTManalysis using ground-based radiometric meadurements (lidar and ODS) and comparison with satellite retrievals ; -retrieval of surface temperature fromsatellite and comparison to in situ measurements ; for validation and identification of possible biases.

+ =

In the RTM analyses snow emissivity from the MODIS UCSB emissivity database was used. For the optimal estimation, LARA radiative transfer and retrieval code was used Preliminary studies (see Figure10 ) show a good agreement between in situ IMB surface temperature measurements and surface temperatures retrieved from IASI spectra.

The MOMO code was developed to allow analysis in both visible and IR spectral domains. It is used here as an illustration to study Arctic radiation budget at the ice/snow-atmosphere interface. We showed that MOMO could be a valuable integration tool combining IAOOS in situ measurements, satellite measurements (IASI, AVHRR, ...) and radiative transfer simulations to:

-compare satellite skin surface temperature products or infrared images with in situ measurements using operational and research satellite products for validation ; -check values of critical parameters such as snow/ice emissivity using temperature retrievals ; -infer cloud properties from IAOOS lidar and radiometric measurements.

This will further allow to analyse surface radiation fluxes (SW and LW) contributing to the energy budget at the ice/snow atmosphere interface.

Combination of ground-based, airborne and satellite observations will be applied to better constrain or validate MOMO radiative transfer simulations to estimate radiative forcing and heating rates in the atmosphere, especially for highly variable conditions (clouds and aerosol layers). Close coincidences between IASI spectral and IAOOS temperature profile measurements allow to compare the snow surface temperature derived from IMB (Ice Mass Balance) profile and the surface temperature derived from the IASI spectrum. Surface temperature was obtained from operational retrievals, (Figure 9) and derived from IASI spectra analysis (see Figure 11) using line-free microwindows and by optimal estimation. First comparison of results is given in Figure 10.

Ether web site http://www.pole-ether.fr IPSL-mesocentre computing facilities were used In the analyses presented here (quick-looks and IASI spectra)

Fig. 1

 1 Fig.1 Agreement of LBLRTM (dashed orange)/CGASA (blue) for water vapor spectroscopy within MODIS 31 channel. (a) water vapor transmission. Spectral response function Of MODIS 31 is in red. (b) water vapor absorption OD (optical depth) within MODIS 31 channel. (c) Zoom on a 80 nm wide band for the water vapor spectrum.

Fig. 2

 2 Fig. 2 Results of the comparision of MOMO to RTTOV, for 6 MODIS channels. Plain line : top of the atmosphere Spectral radiance computed by RTTOV, dashe line : top of The atmosphere spectral radiance computed by MOMO. Dotted line : spectral radiance difference MOMO-RTTOV.

Fig. 3

 3 Fig.3 Results of the comparison of MOMO to CALIPSO-IIR observations. (a) Top of atmosphere brightness temperature observed by Calipso-IIR (green), computed with MOMO (blue), computed by SPIRS operational code (red) for CALIPSO Channels 1-3 (from top to bottom). (b) Differences between the observed and the modelled (in blue with MOMO, in red with SPIRS) brightness temperature at the top of the Atmosphere for CALIPSO channels 1-3.

Fig. 4

 4 Fig. 4 Drift example of a IAOOS buoy (trajectory in red).

Figure 11 :

 11 Figure 11 : Comparison between measured (blueline) and calculated (in red) Brightness Temperature at the Top-of-theatmosphere