Solutions for linear conservation laws with gradient constraints

Several works have developed solutions $u=u(x,t)$ to the linear equation of first order

$$\partial_{t}u+\boldsymbol{b}\cdot\nabla u+cu=f,$$

(1)

for $t>0$ and $x$ in an open subset $\Omega$ of $\mathbb{R}^{N}$, where $\boldsymbol{b}=\boldsymbol{b}(x,t)$ is a given vector field and $c=c(x,t)$ and $f=f(x,t)$ are given functions.

The well-known DiPerna and Lions theory of renormalized solutions, when $\boldsymbol{b}$ is given in Sobolev spaces, has been extended by Ambrosio to BV coefficients for the Cauchy problem and has found several applications in the study of hyperbolic systems of multidimensional conservation laws (see, for instance [1], for an introduction and references). The initial-boundary value problem for (1) with a $C^{1}$ vector field $\boldsymbol{b}$ has been studied in the pioneer work of Bardos [2] using essentially a $L^{2}$ approach for the transport operator. This method also holds for Lipschitz vector fields, as observed in [8], and was extended by Boyer [5] for solenoidal vector fields in Sobolev spaces that do not need to be tangential to the boundary of $\Omega$, i.e. $\boldsymbol{b}\cdot\boldsymbol{n}\neq 0$ on $\partial\Omega$ for $t>0$.

The delicate point is then to prescribe the boundary data to the normal trace of $\boldsymbol{b}$ on the portion of the space-time boundary $\Gamma_{-}\subset\partial\Omega\times(0,T)$ where the characteristics are entering the domain $Q_{T}=\Omega\times(0,T)$. In the case when $\Gamma_{-}$ does not vary with $t$, Besson and Pousin [3] have treated the initial-inflow problems for the continuity equation (1) with $\boldsymbol{L}^{\infty}$ velocity fields $\boldsymbol{b}$ with $c=\nabla\cdot\boldsymbol{b}=\text{div}\,\boldsymbol{b}$ also in $L^{\infty}(Q_{T})$. Recently Crippa et al. [7] have also considered this problem without that restriction on $\Gamma_{-}$ and with similar assumptions on $\boldsymbol{b}$ in BV.

Here we are interested in the initial-boundary value problem for (1) under the additional gradient constraint

$$|\nabla u(x,t)|\leq g(x,t),\quad(x,t)\in Q_{T},$$

(2)

where $g=g(x,t)$ is a given strictly positive and bounded function. This problem was already considered in [20] in the framework of a quasilinear continuity equation

$$\partial_{t}u+\nabla\cdot\boldsymbol{\Phi}(u)=F(u)$$

(3)

and a Lipschitz semilinear lower order term $F=F(x,t,u)$, with a gradient bound in (2) that may depend also on the solution but not on time. As observed in [20], in the linear transport equation (1), corresponding to

$$\boldsymbol{\Phi}(u)=\boldsymbol{b}u\quad\text{and}\quad F(u)=f+\big{(}\nabla\cdot\boldsymbol{b}-c\big{)}u$$

with regular coefficients and $g=g(x)$ independent of $t$, the problem is well-posed in terms of a first order variational inequality with the convex set

$$\mathbb{K}_{g}=\big{\{}v\in H^{1}_{0}(\Omega):|\nabla v(x)|\leq g(x)\text{ a.e. }x\in\Omega\big{\}}.$$

(4)

In [20] it is also proved the existence and asymptotic behaviour of quasivariational solutions for positive nonlinear gradient constraints $g=g(x,u)$ depending continuously on the solution $u=u(x,t)$. Here $H^{1}_{0}(\Omega)$ denotes the usual Sobolev space of functions vanishing on the boundary $\partial\Omega$, as the gradient bound allows to prescribe values on the whole boundary. Moreover, it allows also to consider the data $\boldsymbol{b}$, $c$ and $f$ only in $L^{2}(Q_{T})$, provided $c-\frac{1}{2}\nabla\cdot\boldsymbol{b}$ is bounded from below.

A motivation for the constraint (2) applied to the equation (1) is the “transported sandpile” problem. Following Prigozhin [14, 15], the gradient of the shape of a growing pile of grains $z=z(x,t)$ characterized by its angle of repose $\alpha>0$ is constrained by its surface slope, i.e. $g=\arctan\alpha$. A general conservation of mass, in the form (3) with $\boldsymbol{\Phi}=-\mu\nabla u+\boldsymbol{b}u$ and source density $F$, with transport directed by $\boldsymbol{b}$ and dropping flow directed to the steepest descent $-\mu\nabla u$, should be then subjected to the unilateral conditions

$$\mu\geq 0,\quad|\nabla u|\leq g\quad\text{and}\quad|\nabla u|<g\Rightarrow\mu=0.$$

We illustrate this problem with the interesting example of the one dimensional special case announced in [19]: $\Omega=(0,1)$, $\boldsymbol{b}=1=g$, i.e. $\alpha=\frac{\pi}{4}$ and $f(x,t)=t$. Taking as initial condition the parabola $z_{0}(x)=-\frac{1}{2}x^{2}$, up to the point $\xi_{0}=\sqrt{3}-1$, and the straight line $z_{0}(x)=x-1$, for $\xi_{0}\leq x\leq 1$, the profile of the “transported sandpile” growth attains a steady state exactly at $t=\frac{5}{4}$. This happens with the first free boundary point $\xi(t)$ increasing from $\xi_{0}$ up to $t=\frac{1}{2}$, touching then the boundary $x=1$, and decreasing till the midpoint $x=\frac{1}{2}$. At this point, the free boundary $\xi(t)$ meets a second increasing free boundary $\zeta(t)=2(t-1)$, that appears at $t=1$ and increases up to the final stabilization at $t=\frac{5}{4}.$

The explicit sandpile profile is given by

$$z(x,t)=\left\{\begin{array}[]{ll}tx-\frac{1}{2}x^{2}&\mbox{ if }0\leq x\leq\xi(t)\mbox{ and }0\leq t\leq 1,\vskip 8.53581pt\\ x-1&\mbox{ if }\xi(t)<x\leq 1\mbox{ and }0\leq t\leq\frac{1}{2},\vskip 8.53581pt\\ 1-x&\mbox{ if }\xi(t)<x\leq 1\mbox{ and }\frac{1}{2}<t\leq 1,\vskip 8.53581pt\\ x&\mbox{ if }0\leq x\leq\zeta(t)\mbox{ and }1<t\leq\frac{5}{4},\vskip 8.53581pt\\ tx-\frac{1}{2}x^{2}&\mbox{ if }\zeta(t)<x\leq\xi(t)\mbox{ and }1<t\leq\frac{5}{4},\vskip 8.53581pt\\ x-1&\mbox{ if }\xi(t)<x\leq 1\mbox{ and }1<t\leq\frac{5}{4},\vskip 8.53581pt\\ \frac{1}{2}-|x-\frac{1}{2}|&\mbox{ if }t>\frac{5}{4},\end{array}\right.$$

where $\xi(t)=t-1+\sqrt{(1-t)^{2}+2}$, if $0\leq t\leq\frac{1}{2}$, and $\xi(t)=t+1-\sqrt{(t+1)^{2}-2}$, if $\frac{1}{2}<t\leq\frac{5}{4}$.

It is clear that $z(t)\in\mathbb{K}_{1}\subset H^{1}_{0}(\Omega)$.

We introduce the function $d(x)=\frac{1}{2}-|x-\frac{1}{2}|$ and the convex set

$$\mathbb{K}_{\vee}^{\wedge}=\big{\{}v\in H^{1}_{0}(\Omega):-d(x)\leq v(x)\leq d(x)\text{ a.e. }x\in(0,1)\big{\}}\supset\mathbb{K}_{1}.$$

Since $\partial_{t}z+\partial_{x}z=t$ in $A=\big{\{}(x,t)\in Q_{T}:|\partial_{x}z(x,t)|<1\big{\}}$, by simple computation and integration in $Q_{T}$, we easily conclude that $z$, which (using $\vee=sup$ and $\wedge=inf$) can be written as

$$z(x,t)=\big{(}-d(x)\big{)}\vee\big{(}(tx-\frac{1}{2}x^{2})\wedge d(x)\big{)},$$

is then the unique solution in $\mathbb{K}_{\vee}^{\wedge}$ of the variational inequality

$$\int_{Q_{T}}\big{(}\partial_{t}u+\partial_{x}u-t\big{)}(w-u)\geq 0\quad\forall\,w(t)\in\mathbb{K}_{\vee}^{\wedge},\,0<t<T,\qquad u(0)=z_{0}.$$

(5)

But since $z_{0},z(t)\in\mathbb{K}_{1}$, $z$ is also the solution of the variational inequality (5) with $w(t)\in\mathbb{K}_{1}\subset\mathbb{K}_{\vee}^{\wedge}$, which has at most one solution also in the convex set $\mathbb{K}_{1}$, defined as in (4) with $g\equiv 1$.

In Section 2 we establish the existence and the uniqueness of the solution of the first order variational inequality associated with the general linear equation (1) in a family of time dependent convex sets with gradient constraints of the type (4) with $g=g(x,t)$. We improve the results of [20] under general square integrability assumptions on the coefficients and on the data, by direct estimates in the parabolic-penalized problem and passage to the limit, first in the penalization parameter $\varepsilon$, and afterwards in the regularization parameter $\delta$. The continuous dependence of the solution with respect to the gradient constraint variations in $L^{\infty}$, to the coefficients of the operator and the data in $L^{1}$, is proven in Section 3 under the weak coercive condition (7), as well as the asymptotic convergence towards the unique stationary solution under the stronger coercive assumption (23).

Finaly, in Section 4, we consider the special case of a constant vector $\boldsymbol{b}$, with $g=1$ and $f=f(t)$ bounded, to show the equivalence of the variational inequalities with the gradient constraint and with the two obstacles, i.e. with the signed distances to the boundary constraints on the solution. This is a first result of this type for first order variational inequalities, similar to the elliptic well-known case of the elastoplastic torsion problem (see, for instance, [16] and its references) and to the parabolic case without convection considered in [21, 22], where it was shown that this equivalence is not always possible in the general case. With additional conditions, that include the above one dimensional transported sand pile problem, we establish the finite time stabilization of the solution. This extends to the convective problem a similar result by Cannarsa et al. [6] and raises the interesting open question of establishing more general conditions on the finite time stabilization of evolutionary problems with gradient constraints.

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with a Lipschitz boundary $\partial\Omega$ and, for any $T>0$, denote $Q_{T}=\Omega\times(0,T)$.

Assume that

$$\boldsymbol{b}\in\boldsymbol{L}^{2}(Q_{T})\quad\text{and}\quad c\in L^{2}(Q_{T}),$$

(6)

and there exists $l\in\mathbb{R}$ such that

$$c-\frac{1}{2}\nabla\cdot\boldsymbol{b}\geq l\quad\text{in}\quad Q_{T},$$

(7)

being this inequality satisfied in the distributional sense, since $\nabla\cdot\boldsymbol{b}$ does not need to be a function.

In addition we also suppose given

$$f\in L^{2}(Q_{T})\quad\text{and}\quad u_{0}\in\mathbb{K}_{g(0)},$$

(8)

with

$$g\in W^{1,\infty}\big{(}0,T;L^{\infty}(\Omega)\big{)},\quad g\geq m>0.$$

(9)

As in (4), we define, for $t\geq 0,$

$$\mathbb{K}_{g(t)}=\big{\{}v\in H^{1}_{0}(\Omega):|\nabla v(x)|\leq g(x,t)\mbox{ for a.e. in }x\in\Omega\big{\}}.$$

Consider the following variational inequality problem: To find $u$, in an appropriate space, such that

$$\begin{array}[]{l}\left\{\begin{array}[]{l}u(t)\in\mathbb{K}_{g(t)}\mbox{ for a.e. }t\in(0,T),\qquad u(0)=u_{0},\\ \\ \displaystyle\int_{\Omega}\partial_{t}u(t)(v-u(t))+\displaystyle\int_{\Omega}\boldsymbol{b}(t)\cdot\nabla u(t)(v-u(t))+\displaystyle\int_{\Omega}c(t)\,u(t)(v-u(t))\geq\displaystyle\int_{\Omega}f(t)(v-u(t)),\\ \\ \hfill{\forall\,v\in\mathbb{K}_{g(t)},\mbox{ for a.e. }t\in(0,T)}.\end{array}\right.\end{array}$$

(10)

To prove the existence of a solution, we consider a family of approximating quasilinear parabolic problems for $u^{\varepsilon\delta}$, with $\varepsilon,\delta\in(0,1)$, defined as follows

$$\left\{\begin{array}[]{l}\partial_{t}u^{\varepsilon\delta}-\delta\nabla\cdot(k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\nabla u^{\varepsilon\delta})+\boldsymbol{b}^{\delta}\cdot\nabla u^{\varepsilon\delta}+c^{\delta}\,u^{\varepsilon\delta}=f^{\delta}\mbox{ in }Q_{T},\vskip 8.53581pt\\ u^{\varepsilon\delta}=0\mbox{ on }\partial\Omega\times(0,T),\vskip 8.53581pt\\ u^{\varepsilon\delta}(0)={u_{0}^{\varepsilon}}\mbox{ in }\Omega_{0},\end{array}\right.$$

(11)

where $\boldsymbol{b}^{\delta}$, $c^{\delta}$, $f^{\delta}$ and $u_{0}^{\varepsilon}$ are $\mathscr{C}^{\infty}$ appropriate regularizations of $\boldsymbol{b}$, $c$, $f$ and $u_{0}$, respectively, with $|\nabla u_{0}^{\varepsilon}|\leq g(0)$ and $k_{\varepsilon}$ is a smooth real function such that $k_{\varepsilon}(s)=1$ if $s\leq 0$ and $k_{\varepsilon}(s)=e^{\frac{s}{\varepsilon}}$ if $s\geq\varepsilon$. Notice that this problem has a unique solution $u^{\varepsilon\delta}\in H^{1}\big{(}0,T;L^{2}(\Omega)\big{)}\cap L^{\infty}\big{(}0,T;H^{1}_{0}(\Omega)\big{)}\cap\mathscr{C}(\overline{Q}_{T})$, by the classical theory of parabolic quasilinear problems (see, for instance, [10]).

We prove first several a priori estimates.

Multiplying the equation of the problem (11) by $u^{\varepsilon\delta}$ and integrating over $Q_{t}=\Omega\times]0,t[$, we have

$$\frac{1}{2}\int_{\Omega}|u^{\varepsilon\delta}(t)|^{2}+\delta\int_{Q_{t}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})|\nabla u^{\varepsilon\delta}|^{2}+\int_{Q_{t}}\big{(}\boldsymbol{b}^{\delta}\cdot\nabla u^{\varepsilon\delta}\big{)}\,u^{\varepsilon\delta}+\int_{Q_{t}}c^{\delta}\,|u^{\varepsilon\delta}|^{2}=\int_{Q_{t}}f^{\delta}u^{\varepsilon\delta}+\frac{1}{2}\int_{\Omega}|{u_{0}^{\varepsilon}}|^{2}.$$

Observing that

$$\int_{Q_{t}}\big{(}\boldsymbol{b}^{\delta}\cdot\nabla u^{\varepsilon\delta}\big{)}\,u^{\varepsilon\delta}=-\frac{1}{2}\int_{Q_{t}}\big{(}\nabla\cdot\boldsymbol{b}^{\delta}\big{)}\,\big{(}u^{\varepsilon\delta}\big{)}^{2},$$

and using the coercive inequality for the regularized coefficients

$$c^{\delta}-\tfrac{1}{2}\nabla\cdot\boldsymbol{b}^{\delta}=(c-\tfrac{1}{2}\nabla\cdot\boldsymbol{b})*\rho_{\delta}\geq l*\rho_{\delta}=l,$$

we have

$$\frac{1}{2}\int_{\Omega}|u^{\varepsilon\delta}(t)|^{2}+\delta\int_{Q_{t}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})|\nabla u^{\varepsilon\delta}|^{2}\leq\|f^{\delta}\|_{L^{2}(Q_{T})}\|u^{\varepsilon\delta}\|_{L^{2}(Q_{t})}+\frac{1}{2}\|u_{0}\|_{L^{2}(\Omega)}^{2}+|l|\|u^{\varepsilon\delta}\|_{L^{2}(Q_{t})}^{2}.$$

Hence, by the integral Gronwall’s inequality, there exists a positive constant $C_{T}$, independent of $\varepsilon$ and $\delta$, such that

$$\|u^{\varepsilon\delta}\|_{L^{2}(Q_{T})}\leq C_{T}$$

and so

$$\delta\int_{Q_{t}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})|\nabla u^{\varepsilon\delta}|^{2}\leq C^{\prime},$$

where $C^{\prime}=C^{\prime}(\|f\|_{L^{2}(Q_{T})},\|u_{0}\|_{L^{2}(\Omega))},|l|)$.

On the other hand, we observe that

$$\int_{Q_{T}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})|\nabla u^{\varepsilon\delta}|^{2}=\int_{Q_{T}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\big{(}|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\big{)}+\int_{Q_{T}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})g^{2}.$$

(13)

Since $k_{\varepsilon}(s)=1$ for $s\leq 0$ and $k_{\varepsilon}(s)s\geq 0$, for all $s\geq 0$, then

$$\int_{Q_{t}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\big{(}|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\big{)}=\int_{\{|\nabla u^{\varepsilon\delta}|^{2}\leq g^{2}\}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\big{(}|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\big{)}\\ +\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}\}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\big{(}|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\big{)}\geq-\int_{Q_{T}}g^{2}.$$

(14)

From (13) and (14) we obtain

$$\int_{Q_{t}}k_{\varepsilon}(|\nabla w|^{2}-g^{2})\leq\frac{1}{m^{2}}\left(\int_{Q_{T}}k_{\varepsilon}(|\nabla w|^{2}-g^{2})|\nabla w|^{2}+\int_{Q_{T}}g^{2}\right)\leq\frac{1}{m^{2}}\Big{(}\frac{1}{\delta}C^{\prime}+\|g\|^{2}_{L^{2}(Q_{T})}\Big{)}\leq\frac{1}{\delta}C_{1},$$

where $C_{1}=C_{1}(m,\|f\|_{L^{2}(Q_{T})},\|g\|_{L^{2}(Q_{T})},\|u_{0}\|_{L^{2}(\Omega)},|l|)$.

From (12) we know that

$$\int_{Q_{T}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\leq\frac{1}{\delta}C_{1},$$

where $C_{1}$ is the positive constant of Estimate 1. So,

$$\frac{1}{\delta}C_{1}\geq\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})=\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}e^{\frac{|\nabla u^{\varepsilon\delta}|^{2}-g^{2}}{\varepsilon}}$$

and recalling that, for all $s>0$ and all $j\in\mathbb{N}$, $e^{s}\geq\frac{s^{j}}{j!}$, we get, for any $j\in\mathbb{N}$,

$$\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}\left(|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\right)^{j}\leq j!\varepsilon^{j}\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}e^{\frac{|\nabla u^{\varepsilon\delta}|^{2}-g^{2}}{\varepsilon}}\leq\frac{1}{\delta}j!\varepsilon^{j}C_{1}.$$

Given $1\leq p<\infty$, we have

$$\int_{Q_{T}}|\nabla u^{\varepsilon\delta}|^{2p}=\int_{\{|\nabla u^{\varepsilon\delta}|^{2}\leq g^{2}+\varepsilon\}}|\nabla u^{\varepsilon\delta}|^{2p}+\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}|\nabla u^{\varepsilon\delta}|^{2p}$$

(16)

and, since $g$ is bounded, we can estimate, for any $p\in\mathbb{N}$, the second integral in the second term of (16) as follows,

$$\displaystyle\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}|\nabla u^{\varepsilon\delta}|^{2p}\leq\displaystyle{\int_{\{|\nabla u^{\varepsilon\delta}|^{2}>g^{2}+\varepsilon\}}\sum_{j=0}^{p}}\binom{p}{j}\displaystyle{\|g\|_{L^{\infty}(Q_{T})}^{2p-2j}\left(|\nabla u^{\varepsilon\delta}|^{2}-g^{2}\right)^{j}}\\ \displaystyle{\leq\frac{1}{\delta}\sum_{j=0}^{p}}\binom{p}{j}\|g\|_{L^{\infty}(Q_{T})}^{2p-2j}j!\varepsilon^{j}C_{1}.$$

The first integral in the second term of (16) is clearly bounded since

$$\int_{\{|\nabla u^{\varepsilon\delta}|^{2}\leq g^{2}+\varepsilon\}}|\nabla u^{\varepsilon\delta}|^{2p}\leq\int_{Q_{T}}\left(g^{2}+1\right)^{p}$$

and the conclusion follows easily, first for $2p\in\mathbb{N}$ and afterwards for any $1\leq p<\infty$.

We multiply the equation of problem (11) by $\partial_{t}u^{\varepsilon\delta}$ and we integrate over $Q_{t}$, noting that $\partial_{t}u^{\varepsilon\delta}=0$ on $\partial\Omega\times(0,T)$. Denoting $\phi_{\varepsilon}(s)=\displaystyle{\int_{0}^{s}k_{\varepsilon}(\tau)d\tau}$, we have

$$\int_{Q_{t}}|\partial_{t}u^{\varepsilon\delta}|^{2}+\frac{\delta}{2}\int_{Q_{t}}\frac{d\ }{dt}\left(\phi_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})\right)+\delta\int_{Q_{t}}k_{\varepsilon}(|\nabla u^{\varepsilon\delta}|^{2}-g^{2})g\partial_{t}g\\ +\int_{Q_{t}}(\boldsymbol{b}^{\delta}\cdot\nabla u^{\varepsilon\delta})\,\partial_{t}u^{\varepsilon\delta}+\int_{Q_{t}}c^{\delta}\,u^{\varepsilon\delta}\,\partial_{t}u^{\varepsilon\delta}=\int_{Q_{t}}f^{\delta}\partial_{t}u^{\varepsilon\delta}.$$

We choose $2<s<\frac{2N}{N-2}$ and $q=\frac{2s}{s-2}$, and so we have $\frac{1}{s}+\frac{1}{q}+\frac{1}{2}=1$. Then

$$\Big{|}\int_{Q_{t}}(\boldsymbol{b}^{\delta}\cdot\nabla u^{\varepsilon\delta})\,\partial_{t}u^{\varepsilon\delta}\Big{|}\leq\|\boldsymbol{b}^{\delta}\|_{L^{s}(Q_{T})}\|\nabla u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}\|\partial_{t}u^{\varepsilon\delta}\|_{L^{2}(Q_{T})}\leq\|\boldsymbol{b}^{\delta}\|_{L^{s}(Q_{T})}^{2}\|\nabla u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}^{2}+\frac{1}{4}\|\partial_{t}u^{\varepsilon\delta}\|^{2}_{L^{2}(Q_{T})},$$

and

$$\Big{|}\int_{Q_{t}}c^{\delta}\,u^{\varepsilon\delta}\,\partial_{t}u^{\varepsilon\delta}\Big{|}\leq\|c^{\delta}\|_{L^{s}(Q_{T})}\|u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}\|\partial_{t}u^{\varepsilon\delta}\|_{L^{2}(Q_{T})}\leq\|c^{\delta}\|_{L^{s}(Q_{T})}^{2}\|u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}^{2}+\frac{1}{4}\|\partial_{t}u^{\varepsilon\delta}\|^{2}_{L^{2}(Q_{T})}.$$

So

	$\displaystyle\frac{1}{4}\int_{Q_{T}}|\partial_{t}u^{\varepsilon\delta}|^{2}\leq$	$\displaystyle\|f^{\delta}\|_{L^{2}(Q_{T})}^{2}+\frac{\delta}{2}\int_{\Omega}\phi_{\varepsilon}\big{(}|\nabla u^{\varepsilon\delta}(0)|^{2}-g^{2}(0)\big{)}-\frac{\delta}{2}\int_{\Omega}\phi_{\varepsilon}\big{(}|\nabla u^{\varepsilon\delta}(t)|^{2}-g^{2}(t)\big{)}$
		$\displaystyle+C_{1}\|g\|_{L^{\infty}(Q_{T})}\|\partial_{t}g\|_{L^{\infty}(Q_{T})}+\big{(}\|\boldsymbol{b}^{\delta}\|_{L^{s}(Q_{T})}^{2}+C_{q}\|c^{\delta}\|_{L^{s}(Q_{T})}^{2}\big{)}\|\nabla u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}^{2},$

being $C_{q}$ a Poincaré constant. Observe that, since $\Omega$ is bounded we may find a positive upper bound $C_{3}$ of $C_{q}$, independently of $q\leq\infty$.

On one hand

$$\displaystyle\int_{\Omega}\phi_{\varepsilon}\big{(}|\nabla u^{\varepsilon\delta}(0)|^{2}-g^{2}(0)\big{)}\leq 0,$$

because $|\nabla u^{\varepsilon\delta}(0)|=|\nabla{u_{0}^{\varepsilon}}|\leq g(0)$. On the other hand, if we set $\Lambda=\{(x,t)\in Q_{T}:|\nabla u^{\varepsilon\delta}(x,t)|<g(x,t)\}$, we have

	$\displaystyle\qquad\phi_{\varepsilon}(|\nabla u^{\varepsilon\delta}(x,t)|^{2}-g^{2}(x,t))=|\nabla u^{\varepsilon\delta}(x,t)|^{2}-g^{2}(x,t)\geq-g^{2}(x,t)\qquad\mbox{for a.e. }(x,t)\in\Lambda,$
	$\displaystyle\qquad\phi_{\varepsilon}(|\nabla u^{\varepsilon\delta}(x,t)|^{2}-g^{2}(x,t))\geq 0\geq-g^{2}(x,t)\qquad\mbox{for a.e. }(x,t)\in Q_{T}\setminus\Lambda.$

Consequently, for a.e. $t\in(0,T)$,

$$-\int_{\Omega}\phi_{\varepsilon}(|\nabla u^{\varepsilon\delta}(t)|^{2}-g^{2}(t))\leq\|g\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}.$$

So,

$$\frac{1}{4}\|\partial_{t}u^{\varepsilon\delta}\big{\|}^{2}_{L^{2}(Q_{T})}\leq\|f^{\delta}\|^{2}_{L^{2}(Q_{T})}+\frac{\delta}{2}\|g\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}\\ +C_{1}\|g\|_{L^{\infty}(Q_{T})}\|\partial_{t}g\|_{L^{\infty}(Q_{T})}+\big{(}\|\boldsymbol{b}^{\delta}\|_{L^{s}(Q_{T})}^{2}+C_{q}\|c^{\delta}\|_{L^{s}(Q_{T})}^{2}\big{)}\|\nabla u^{\varepsilon\delta}\|_{L^{q}(Q_{T})}^{2},$$

and the proof of Estimate 3 is concluded.

By (15) and (17), we know there exist constants $D_{\delta}$, $C_{\delta}$ and $C_{4}$, independent of $\varepsilon$, such that, for each $N<p<\infty$,

$$\|u^{\varepsilon\delta}\|_{L^{p}(0,T;W^{1,p}_{0}(\Omega))}\leq D_{\delta},\qquad\|\partial_{t}u^{\varepsilon\delta}\|_{L^{2}(Q_{T})}\leq\big{(}\|\boldsymbol{b}^{\delta}\|_{\boldsymbol{L}^{s}(Q_{T})}+\|c^{\delta}\|_{L^{s}(Q_{T})}\big{)}C_{\delta}+C_{4}.$$