Consider the following elliptic equation of divergence form in $ {{\mathbb R}}^d $ ($ d{\geqslant} 2 $):

$ \begin{align} {\mathord{{{\rm{div}}}}} (a\cdot\nabla u) = 0, \end{align} $

(1)

where $ a:{{\mathbb R}}^d\to {{\mathbb R}}^{d\times d} $ is a Borel measurable function and $ \nabla: = ({\partial}_{x_1},\cdots,{\partial}_{x_d}) $. When $ a $ is uniformly elliptic, the celebrated works of De-Giorgi [1] and Nash [2] said that any weak solutions of elliptic equation (1) are bounded and Hölder continuous. Moreover, Moser [3] showed that any weak solutions of (1) satisfy the Harnack inequality. In [4], Trudinger considered the non-uniformly elliptic equation (1) under the following integrability assumptions:

$ \lambda^{-1}_0\in L^{p_0},\ \mu_0\in L^{p_1}\mbox{ with $p_0,p_1\in(1,\infty]$ satisfying $\tfrac{1}{p_0}+\tfrac{1}{p_1}<\tfrac{2}{d}$,} $

where

$ \begin{align} \lambda_0(x): = \inf\limits_{|\xi| = 1}\xi\cdot a(x)\xi,\quad \mu_0(x): = \sup\limits_{|\xi| = 1}\frac{|a(x)\xi|^2}{\xi\cdot a(x)\xi}. \end{align} $

(2)

He showed that any generalized solutions of (1) are locally bounded and weak Harnack inequality holds. Recently, Bella and Schäffner [5] showed the same results under the following sharp condition on $ p_0, p_1 $,

$ \begin{align} \tfrac{1}{p_0}+\tfrac{1}{p_1}<\tfrac{2}{d-1},\ \ p_0,p_1\in[1,\infty], \end{align} $

(3)

Here we extend the main result of [5] to parabolic case. More precisely, we consider the following parabolic equation of divergence form in $ {{\mathbb R}}^{d+1} $:

$ \begin{align} {\partial}_t u = {\mathord{{{\rm{div}}}}} (a\cdot\nabla u)+b\cdot \nabla u+f, \end{align} $

(4)

where

$ a:{{\mathbb R}}^{d+1}\to{{\mathbb R}}^{d\times d},\ b:{{\mathbb R}}^{d+1}\to{{\mathbb R}}^d,\ f:{{\mathbb R}}^{d+1}\to{{\mathbb R}} $

are Borel measurable functions. As in (2), we introduce

$ \begin{align} \lambda(x): = \inf\limits_{t{\geqslant} 0, |\xi| = 1}\xi\cdot a(t,x)\xi,\quad \mu(x): = \sup\limits_{t{\geqslant} 0, |\xi| = 1}\frac{|a(t,x)\xi|^2}{\xi\cdot a(t,x)\xi}, \end{align} $

(5)

and suppose that $ \lambda $ and $ \mu $ are nonnegative Borel measurable functions.

Definition 0.1  A continuous function $ u:{{\mathbb R}}^{d+1}\to{{\mathbb R}} $ is called a Lipschitz weak (super/sub)-solution of PDE (4) if $ \nabla u $ is locally bounded and for any nonnegative Lipschitz function $ \varphi $ on $ {{\mathbb R}}^{d+1} $ with compact support,

$ \begin{align} -《u,{\partial}_t\varphi》 = ({\geqslant}/{\leqslant})-《a\cdot\nabla u,\nabla\varphi>\!\!+《b\cdot\nabla u,\varphi》+《f,\varphi》, \end{align} $

(6)

where $ 《f,g》: = \int_{{\mathbb R}}\int_{{{\mathbb R}}^{d}}f(t,x)g(t,x){{\mathord{{{\rm{d}}}}}} x{{\mathord{{{\rm{d}}}}}} t $.

For $ p,q\in[1,\infty] $, let $ {{\mathbb L}}^{q,p}_{t,x}: = L^q({{\mathbb R}}; L^p({{\mathbb R}}^d)) $ and $ {{\mathbb L}}^{p,q}_{x,t}: = L^p({{\mathbb R}}^d; L^q({{\mathbb R}})) $ be the space of space-time functions with norms, respectively,

$ \|f\|_{{{\mathbb L}}^{q,p}_{t,x}}: = \left(\int_{{{\mathbb R}}}\|f(t,\cdot)\|_p^q{{\mathord{{{\rm{d}}}}}} t\right)^{1/q},\ \|f\|_{{{\mathbb L}}^{p,q}_{x,t}}: = \left(\int_{{{\mathbb R}}^d}\|f(\cdot,x)\|_q^p{{\mathord{{{\rm{d}}}}}} x\right)^{1/p}, $

where $ \|\cdot\|_p $ stands for the usual $ L^p $-norm. For $ r>0 $ and $ (s,z)\in{{\mathbb R}}^{d+1} $, we define

$ Q_r: = [-r^2,r^2]\times B_r\subset{{\mathbb R}}^{d+1},\ Q^{s,z}_r: = Q_r+(s,z),\ \ B^z_r: = B_r+z, $

and for $ p\in[1,\infty] $, introduce the following localized $ L^p $-space:

$ \begin{align} {\widetilde L}^p: = \Big\{f\in L^1_{loc}({{\mathbb R}}^d): {|\mspace{-3mu}|\mspace{-3mu}|} f{|\mspace{-3mu}|\mspace{-3mu}|}_p: = \sup\limits_z\|{{\bf{1}}}_{B^z_1}f\|_p<\infty\Big\}, \end{align} $

(7)

and for $ p,q\in[1,\infty] $,

$ \begin{align} \widetilde{{\mathbb L}}^{q,p}_{t,x}: = \Big\{f\in L^1_{loc}({{\mathbb R}}^{d+1}): {|\mspace{-3mu}|\mspace{-3mu}|} f{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q,p}_{t,x}}: = \sup\limits_{s,z}\|{{\bf{1}}}_{Q^{s,z}_1}f\|_{{{\mathbb L}}^{q,p}_{t,x}}<\infty\Big\}, \end{align} $

(8)

and similarly for $ \widetilde{{\mathbb L}}^{p,q}_{x,t} $.

Below we fix $ p_0\in(\frac{d}{2},\infty] $ and $ p_1\in[1,\infty] $ with

$ \begin{align} \tfrac1{p_0}+\tfrac{1}{p_1}<\tfrac{2}{d-1}, \end{align} $

(9)

and introduce the index set

$ {{\mathbb I}}^d_{p_0}: = \Big\{(p,q)\in[1,\infty]^2: \tfrac{1}{p}<(1-\tfrac{1}{q})(\tfrac{2}{d}-\tfrac{1}{p_0})\Big\}. $

We make the following assumptions about $ a $ and $ b $:

(H$ ^a $) $ {|\mspace{-3mu}|\mspace{-3mu}|}\lambda^{-1}{|\mspace{-3mu}|\mspace{-3mu}|}_{p_0}+{|\mspace{-3mu}|\mspace{-3mu}|}\mu{|\mspace{-3mu}|\mspace{-3mu}|}_{p_1}<\infty $, where $ \lambda,\mu $ are defined by (5).

(H$ ^b $) $ b = b_1+b_2 $, where if $ p_0\in(\frac d2,d] $, $ b_1\equiv 0 $, and if $ p_0>d $, $ b_1\in\widetilde{{\mathbb L}}^{q_2,p_2}_{t,x} $ for some $ (p_2,q_2)\in[1,\infty]^2 $ with

$ \begin{align} \tfrac{1}{2p_0}+\tfrac{1}{p_2}<(\tfrac12-\tfrac{1}{q_2})(\tfrac{2}{d}-\tfrac{1}{p_0}), \end{align} $

(10)

and $ b_2\in\widetilde{{\mathbb L}}^{p_1,\infty}_{x,t} $ and $ {\mathord{{{\rm{div}}}}} b_2\equiv 0 $.

For simplicity of notations, we introduce the following parameter set

$ \begin{align} \Theta: = \Big(d,p_i, q_i,{|\mspace{-3mu}|\mspace{-3mu}|}\lambda^{-1}{|\mspace{-3mu}|\mspace{-3mu}|}_{{p_0}},{|\mspace{-3mu}|\mspace{-3mu}|}\mu{|\mspace{-3mu}|\mspace{-3mu}|}_{p_1}, {|\mspace{-3mu}|\mspace{-3mu}|} b_1{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q_2,p_2}_{t,x}}, {|\mspace{-3mu}|\mspace{-3mu}|} b_2{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{p_1,\infty}_{x,t}}\Big). \end{align} $

(11)

We have the following apriori estimate.

Theorem 0.2  Under (H$ ^a $) and (H$ ^b $), for any $ f\in\widetilde{{\mathbb L}}^{q_4,p_4}_{t,x} $ with $ (p_4,q_4)\in{{\mathbb I}}^d_{p_0} $ and for any $ T>0 $, there exists a constant $ C = C(T,\Theta, p_4,q_4)>0 $ such that for any Lipschitz weak solution $ u $ of PDE (4) in $ {{\mathbb R}}^{d+1} $ with $ u(t)|_{t{\leqslant} 0}\equiv 0 $,

$ \begin{align} \|u\|_{L^\infty([0,T]\times{{\mathbb R}}^d)}{\leqslant} C{|\mspace{-3mu}|\mspace{-3mu}|} f{{\bf{1}}}_{[0,T]}{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q_4,p_4}_{t,x}}. \end{align} $

(12)

Consider the following heat equation with divergence free drift $ b $:

$ \begin{align} {\partial}_t u = \Delta u+b\cdot \nabla u+f,\ u(t)|_{t{\leqslant} 0} = 0. \end{align} $

(13)

The following apriori global boundedness estimate is a direct consequence of Theorem 0.2.

Corollary 0.3  Let $ b\in\widetilde{{\mathbb L}}^{p,\infty}_{x,t} $ with $ {\mathord{{{\rm{div}}}}} b = 0 $, where $ p\in[1,\infty]\cap(\frac{d-1}{2},\infty] $. For any $ T>0 $ and $ f\in\widetilde{{\mathbb L}}^{q',p'}_{t,x} $, where $ p',q'\in[1,\infty] $ satisfy $ \frac{d}{p'}+\frac{2}{q'}<2 $, there exists a constant $ C>0 $ only depending on $ T,d,p,p',q' $ and $ \|b\|_{\widetilde{{\mathbb L}}^{p,\infty}_{x,t}} $ such that for any Lipschitz weak solution $ u $ of (13),

$ \begin{align} \|u\|_{L^\infty([0,T]\times{{\mathbb R}}^d)}{\leqslant} C{|\mspace{-3mu}|\mspace{-3mu}|} f{{\bf{1}}}_{[0,T]}{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q',p'}_{t,x}}. \end{align} $

(14)

Remark 0.4  Note that when $ \tfrac{d}{p}+\tfrac{2}{q}<2 $ and $ b\in\widetilde{{\mathbb L}}^{q,p}_{t,x} $ with $ {\mathord{{{\rm{div}}}}} b = 0 $, it is well known that (14) holds (cf. [6], [7]). When $ b $ does not depend on $ t $, the current condition $ p>\frac{d-1}{2} $ in Corollary 0.3 is clearly better than $ p>\frac d2 $.

As an application of the global boundedness estimate (12), we consider the following SDE:

$ \begin{align} {{\mathord{{{\rm{d}}}}}} X_t = \sqrt{2}\sigma(t,X_t){{\mathord{{{\rm{d}}}}}} W_t+b(t,X_t){{\mathord{{{\rm{d}}}}}} t,\ \ X_0 = x, \end{align} $

(15)

where $ W $ is a $ d $-dimensional standard Brownian motion. We recall the following notion of weak solutions to SDE (15).

Definition 0.5  Let $ \frak{F}: = (\Omega,{{\mathscr F}},{{\bf P}}; ({{\mathscr F}}_t)_{t{\geqslant} 0}) $ be a stochastic basis and $ (X,W) $ a pair of $ {{\mathscr F}}_t $-adapted processes defined thereon. We call triple $ (\frak{F}, X,W) $ a weak solution of SDE (15) with starting point $ x\in{{\mathbb R}}^d $ if

(i)  $ {{\bf P}}(X_0 = x) = 1 $ and $ W $ is an $ {{\mathscr F}}_t $-Brownian motion;

(ii)  for all $ t{\geqslant} 0 $, it holds that $ {{\bf P}} $-a.s.

$ \int^t_0\Big(|\sigma(s,X_s)|^2+|b(s,X_s)|\Big){{\mathord{{{\rm{d}}}}}} s<\infty,\ \ a.s., \text{and}\, X_t = x+\sqrt{2}\int^t_0\sigma(s,X_s){{\mathord{{{\rm{d}}}}}} W_s+\int^t_0b(s,X_s){{\mathord{{{\rm{d}}}}}} s. $

We make the following assumptions about $ \sigma $ and $ b $.

($ \widetilde {\bf H}^\sigma $) Suppose that there are a sequence of $ d\times d $-matrix functions $ \sigma_n\in L^\infty({{\mathbb R}}_+; C^\infty_b) $, $ (p_2,q_2)\in{{\mathbb I}}^d_{p_0} $ and $ \kappa_0>0 $ such that for all $ n\in{{\mathbb N}} $,

$ \begin{align} \,\,\qquad{|\mspace{-3mu}|\mspace{-3mu}|}\lambda^{-1}_n{|\mspace{-3mu}|\mspace{-3mu}|}_{p_0}+{|\mspace{-3mu}|\mspace{-3mu}|}\mu_n{|\mspace{-3mu}|\mspace{-3mu}|}_{p_1}+{|\mspace{-3mu}|\mspace{-3mu}|} {\partial}_ia^{ij}_n{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{p_1,\infty}_{x,t}} +{|\mspace{-3mu}|\mspace{-3mu}|} ({\partial}_i{\partial}_ja^{ij}_n)^+{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q_2,p_2}_{t,x}}{\leqslant}\kappa_0, \end{align} $

(16)

where $ a_n: = \sigma_n\sigma^*_n $, $ \lambda_n $ and $ \mu_n $ are defined as in (5) by $ a_n $. Moreover, for some $ p_3,q_3\in[2,\infty] $ with $ (\frac{p_3}{2},\frac{q_3}{2})\in{{\mathbb I}}^d_{p_0} $ and for any $ T,R>0 $,

$ \begin{align} \qquad\sup\limits_n{|\mspace{-3mu}|\mspace{-3mu}|}\sigma_n{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q_3,p_3}_{t,x}} = :\kappa_1<\infty,\quad \lim\limits_{n\to\infty}\|(\sigma_n-\sigma){{\bf{1}}}_{[0,T]\times B_R}\|_{{{\mathbb L}}^{q_3,p_3}_{t,x}} = 0. \end{align} $

(17)

($ \widetilde {\bf H}^b $) Let $ b = b_1+b_2 $ satisfy ($ {\bf H}^b $) and $ b\in\widetilde{{\mathbb L}}^{q_4,p_4}_{t,x} $ for some $ (p_4,q_4)\in{{\mathbb I}}^d_{p_0} $.

We have the following existence result.

Theorem 0.6  Under ($ \widetilde {\bf H}^\sigma $) and ($ \widetilde {\bf H}^b $), for any $ x\in{{\mathbb R}}^d $, there is at least one weak solution $ (\frak{F}, X,W) $ for SDE (15). Moreover, for any $ (p,q)\in{{\mathbb I}}^d_{p_0} $ and $ T>0 $, there are $ \theta\in(0,1) $ and constant $ C = C(T,\Theta,p,q)>0 $ such that for any stopping time $ \tau{\leqslant} T $, $ \delta\in(0,1) $ and $ f\in\widetilde{{\mathbb L}}^{q,p}_{t,x} $,

$ \begin{align} {{\bf E}}\left(\int^{\tau+\delta}_{\tau}f(s, X_s){{\mathord{{{\rm{d}}}}}} s\Big|{{\mathscr F}}_{\tau}\right){\leqslant} C\delta^\theta{|\mspace{-3mu}|\mspace{-3mu}|} f{|\mspace{-3mu}|\mspace{-3mu}|}_{\widetilde{{\mathbb L}}^{q,p}_{t,x}}. \end{align} $

(18)

The following two examples can be derived from the above existence result.

Example 0.7  Let $ d{\geqslant} 3 $ and $ \alpha\in(0,(\frac d2-1)\wedge(\frac12+\frac{1}{d-1})) $, $ \beta\in(0,2\alpha) $. For any $ \lambda{\geqslant}0 $ and $ x\in{{\mathbb R}}^d $, the following SDE admits a unique strong solution:

$ {{\mathord{{{\rm{d}}}}}} X_t = |X_t|^{-\alpha}{{\mathord{{{\rm{d}}}}}} W_t+\lambda X_t|X_t|^{-\beta-1}{{\mathord{{{\rm{d}}}}}} t,\ \ X_0 = x. $

Note that the starting point can be zero and the uniqueness follows from [8].

Example 0.8  The following two dimensional degenerate SDE admits a solution:

$ \left\{ \begin{aligned} {{\mathord{{{\rm{d}}}}}} X^1_t = |X^2_t|^\alpha{{\mathord{{{\rm{d}}}}}} W^1_t+b^1(X_t){{\mathord{{{\rm{d}}}}}} t,\\ {{\mathord{{{\rm{d}}}}}} X^2_t = |X^1_t|^\alpha{{\mathord{{{\rm{d}}}}}} W^2_t+b^2(X_t){{\mathord{{{\rm{d}}}}}} t, \end{aligned} \right. $

where $ \alpha\in(0,\frac12) $ and $ b = (b^1, b^2)\in\widetilde L^{p}({{\mathbb R}}^2) $ for some $ p>\frac{1}{1-2\alpha} $.

More details can be found in [9].