Incompressible Limit of a Mechanical Model for Tissue Growth with Non-Overlapping Constraint

Abstract

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele-Shaw free boundary problem.

1. Introduction

Mathematical models are now commonly used in the study of growth of cell tissue. For instance, a wide literature is now available on the study of the tumor growth through mathematical modeling and numerical simulations [2, 3, 14, 18]. In such models, we may distinguish two kinds of description: Either they describe the dynamics of cell population density (see e.g. [6, 8]), or they consider the geometric motion of the tissue through a free boundary problem of Hele-Shaw type (see e.g. [16, 15, 11, 18]). Recently the link between both descriptions has been investigated from a mathematical point of view thanks to an incompressible limit [22].

In this paper, we depart from the simplest cell population model as proposed in [7]. In this model the dynamics of the cell density is driven by pressure forces and cell multiplication. More precisely, let us denote by n(t,x) the cell density depending on time t ≥ 0 and position x ∈ ℝ^d, and by p the mechanical pressure. The mechanical pressure depends only on the cell density and is given by a state law p = Π(n). Cell proliferation is modelled by a pressure-limited growth function denoted G. Mechanical pressure generates cells displacement with a velocity whose field $𝜐$ is computed thanks to the Darcy’s law. After normalizing all coefficients, the model reads

$$\begin{array}{l}{\partial }_{t}n+\nabla \cdot (nv)=nG(p),\text{on}{ℝ}^{+}\times {ℝ}^d,\\ v=-\nabla p,p=\Pi (n).\end{array}$$

The choice $\Pi (n)=\frac{\gamma }{\gamma -1}{n}^{\gamma -1}$ has been taken in [22, 23, 24]. This choice allows to recover the well-known porous medium equation for which a lot of nice mathematical properties are now well-established (see e.g. [26]). The incompressible limit is then obtained by letting γ going to +∞.

However, this state law does not prevent cells to overlap. In fact, it is not possible with this choice to avoid the cell density to take value above 1 (which corresponds here to the maximal packing density after normalization). A convenient way to avoid cells overlapping is to consider a pressure law which becomes singular when the cell density approaches 1. Such type of singularity is encountered, for instance, in the kinetic theory of dense gases where the interaction between molecules is strongly repulsive at very short distance [9]. Similar singular pressure laws have been also considered in [12, 13] to model collective motion, in [4, 5] to model the traffic flow, and in [21] to model crowd motion (see also the review article [19]). Then, in order to fit this non-overlapping constraint, we consider the following simple model of pressure law given by

$$P(n)=ϵ\frac{n}{1-n}.$$

Finally, the model under study in this paper reads, for ϵ > 0,

$${\partial }_t{n}_{ϵ}-\nabla \cdot (n_{ϵ}\nabla p_{ϵ})=n_{ϵ}G(p_{ϵ}),$$ (1.1)

$$p_{ϵ}=P(n_{ϵ})=ϵ\frac{n_{ϵ}}{1-n_{ϵ}}.$$ (1.2)

This system is complemented by an initial data denoted $n_{ϵ}^{ini}.$ The aim of this paper is to investigate the incompressible limit of this model, which consists in letting ϵ going to 0 in the latter system.

At this stage, it is of great importance to observe that from (1.1), we may deduce an equation for the pressure by simply multiplying (1.1) by P′(n_ϵ) and using the relation $n_{ϵ}=\frac{p_{ϵ}}{ϵ+p_{ϵ}}$ from (1.2),

$${\partial }_t{p}_{ϵ}-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\Delta p_{ϵ}-|\nabla p_{ϵ}{|}^2=(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ}).$$ (1.3)

Formally, we deduce from (1.3) that when ϵ → 0, we expect to have the relation

$$-p_0^2\Delta p_0=p_0^{2}G(p_0).$$ (1.4)

Moreover, passing formally to the limit into (1.2), it appears clearly that (1 − n₀)p₀ = 0. We deduce from this relation that if we introduce the set Ω₀(t) = {p₀ > 0}, then we obtain a free boundary problem of Hele-Shaw type: On Ω₀(t), we have n₀ = 1 and −Δp₀ = G(p₀), whereas p₀ = 0 on ℝ^d \Ω₀(t). Thus although the pressure law is different, we expect to recover the same free boundary Hele-Shaw model as in [22].

The incompressible limit of the above cell mechanical model for tumor growth with a pressure law given by $\Pi (n)=\frac{\gamma }{\gamma -1}{n}^{\gamma -1}$ has been investigated in [22] and in [23] when taking into account active motion of cells. In [24], the case with viscosity, where the Darcy’s law is replaced by the Brinkman’s law, is studied. We mention also the recent works [17, 20] where the incompressible limit with more general assumptions on the initial data has been investigated. However, in all these mentionned works the pressure law do not prevent the non-overlapping of cells. Up to our knowledge, this work is the first attempt to extend the previous result with this constraint, i.e. with a singular pressure law as given by (1.2).

The outline of the paper is the following. In the next section we give the statement of the main result in Theorem 2.1, which is the convergence when ϵ goes to 0 of the mechanical model (1.1)–(1.2) towards the Hele-Shaw free boundary system. The rest of the paper is devoted to the proof of this result. First, in section 3 we establish some a priori estimate allowing to obtain space compactness. Then, section 4 is devoted to the study of the time compactness. Thanks to compactness results, we can pass to the limit ϵ → 0 in system (1.1)–(1.2) in section 5, up to the extraction of a subsequence. Finally the proof of the complementary relation (1.4) is performed in section 6.

2. Main result

The aim of this paper is to establish the incompressible limit ϵ → 0 of the cell mechanical model with non-overlapping constraint (1.1)–(1.2). Before stating our main result, we list the set of assumptions that we use on the growth fonction and on the initial data. For the growth function, we assume

$$\{\begin{array}{l}\exists G_m>0,\Vert G{\Vert }_{\infty }\le G_m,\hfill \\ G^{\prime }<0,\text{and}\exists \gamma >0,\underset{[0,P_M]}{{\displaystyle \text{min}}}|G^{\prime }|=\gamma ,\hfill \\ \exists P_M>0,G(P_M)=0.\hfill \end{array}$$ (2.1)

The quantity P_M, for which the growth stops, is commonly called the homeostatic pressure [25]. This set of assumptions on the growth function is quite similar to the one in [22], except for the bound on the growth term which is needed here due to the singularity in the pressure law.

For the initial data, we assume that there exists ϵ₀ > 0 such that for all ϵ ∈ (0,ϵ₀),

$$\{\begin{array}{l}0\le n_{ϵ}^{ini},p_{ϵ}^{ini}≔ϵ\frac{n_{ϵ}^{ini}}{ϵ+n_{ϵ}^{ini}}\le P_M,\hfill \\ \Vert {\partial }_{x_i}{n}_{ϵ}^{ini}{\Vert }_{L^1({ℝ}^d)}\le C,i=1,\dots ,d,\hfill \\ \exists n_0^{ini}\in L_{+}^1({ℝ}^d),\Vert n_{ϵ}^{ini}-n_0^{ini}{\Vert }_{L^1({ℝ}^d)}\to 0\text{as}ϵ\to 0,\hfill \\ \exists K\subset {ℝ}^d,K\text{compact},\forall ϵ\in (0,{ϵ}_0),\text{supp}{n}_{ϵ}^{ini}\subset K.\hfill \end{array}$$ (2.2)

Notice that this set of assumptions imply that $n_{ϵ}^{ini}$ is uniformly bounded in W^1,1 (ℝ^d).

We are now in position to state our main result.

Theorem 2.1. Let T > 0, Q_T = (0,T) × ℝ^d. Let G and $(n_{ϵ}^{ini})$ satisfy assumptions (2.1) and (2.2) respectively. After extraction of subsequences, both the density n_ϵ and the pressure p_ϵ converge strongly in L¹(Q_T) as ϵ → 0 to the limit n₀ ∈ C([0,T ];L¹(ℝ^d))∩BV (Q_T) and p₀ ∈ BV (Q_T)∩L²([0,T];H¹(ℝ^d)), which satisfy

$$0\le n_0\le 1,0\le p_0\le P_M,$$ (2.3)

$${\partial }_t{n}_0-\Delta p_0=n_{0}G(p_0),in{𝒟}^{\prime }(Q_T),$$ (2.4)

and

$${\partial }_t{n}_0-\nabla \cdot (n_0\nabla p_0)=n_{0}G(p_0),in{𝒟}^{\prime }(Q_T).$$ (2.5)

Moreover, we have the relation

$$(1-n_0)p_0=0,$$ (2.6)

and the complementary relation

$$p_0^2(\Delta p_0+G(p_0))=0,in{𝒟}^{\prime }(Q_T).$$ (2.7)

This result extends the one in [22] to singular pressure laws with non-overlapping constraint. We notice that we recover the same limit model whose uniqueness has already been stated in [22, Theorem 2.4].

Although our proof follows the idea in [22], several technical difficulties must be overcome due to the singularity of the pressure law. Indeed, we first recall that with the choice $\prod (n)=\frac{\gamma }{\gamma -1}{n}^{\gamma -1},$ equation (1.1) may be rewritten as the porous medium equation ∂_tn + Δn^γ = nG(Π(n)). A lot of estimates are known and well established for this equation (see [26]), in particular a semiconvexity estimate is used in [22] which allows to obtain estimate on the time derivative and thus compactness. With our choice of pressure law, (1.1) should be consider as a fast diffusion equation. Thus we have first to state a comparison principle to obtain a priori estimates (see Lemma 3.2). Unlike in [22], we may not use a semiconvexity estimate to obtain estimate on the time derivative. To do so, we use regularizing effects (see section 4). Then the convergence proof has to be adapted for these new estimates.

Finally, we illustrate the comparison between the two pressure laws P and Π by a numerical simulation. We display in Figure 2.1 the density computed thanks to a discretization with an upwind scheme of (1.1). In Figure 2.1-left, the pressure law is $p=P(n)=ϵ\frac{n}{1-n}$ as in (1.2) with ϵ = 0.5. In Figure 2.1-right, the pressure law is $p=\prod (n)=\frac{\gamma }{\gamma -1}{n}^{\gamma }$ with γ = 20. We take G(p) = 10(10 − p)₊ as growth function (which satisfies obviously assumption (2.1) with P_M = 10). The dashed lines in these plots correspond to the constant value 1. As expected, we observe that the density n is bounded by 1 in the case of the pressure law P whereas it takes values greater than 1 for the pressure law Π. This observation illustrates the fact that the choice of the pressure law Π does not prevent from overlapping.

Figure 2.1.

Comparison between numerical solutions computed with two different pressure laws. The red line correspond to the cell density n solving (1.1), the dashed line correspond to the constant value 1. On the left, the pressure law is $p=P(n)=0.5\frac{n}{1-n}.$ On the right, the pressure law is $p=\prod (n)=\frac{\gamma }{\gamma -1}{n}^{\gamma }$ with γ = 20.

3. A priori estimates

3.1. Nonnegativity principle

The following Lemma establishes the nonnegativity of the density.

Lemma 3.1. Let (n_ϵ,p_ϵ) be a solution to (1.1) such that $n_{ϵ}^{ini}\ge 0$ and ‖G‖_∞ ≤ G_m < ∞. Then, for all t ≥ 0, n_ϵ(t) ≥ 0.

Proof. We have the equation

$${\partial }_t{n}_{ϵ}-\nabla \cdot (n_{ϵ}\nabla p_{ϵ})=n_{ϵ}G(p_{ϵ}).$$

We use the Stampaccchia method. We multiply by 1_nϵ<0, then using the notation |n|₋ = max(0,−n) for the negative part, we get

$$\frac{d}{dt}|n_{ϵ}{|}_{-}-\nabla \cdot (|n_{ϵ}{|}_{-}\nabla p_{ϵ})=|n_{ϵ}{|}_{-}G(p_{ϵ}).$$

We integrate in space, using assumption (2.1), we deduce

$$\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}{|}_{-}}dx\le {\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}{|}_{-}}G(p_{ϵ})dx\le G_m{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}{|}_{-}dx}.$$

So, after a time integration

$${\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}{|}_{-}}dx\le e^{G_{m}t}{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}^{ini}{|}_{-}}dx.$$

With the initial condition $n_{ϵ}^{ini}\ge 0$ we deduce n_ϵ≥0.

3.2. A priori estimates

In order to use compactness results, we need first to find a priori estimates on the pressure and the density. We first observe that we may rewrite system (1.1) as, by using (1.2),

$${\partial }_t{n}_{ϵ}-\Delta H(n_{ϵ})=n_{ϵ}G(P(n_{ϵ})),$$ (3.1)

with $H(n)={\displaystyle {\int }_0^{n}u{P}^{\prime }(u)du=P(n)-ϵ\text{ln}(P(n)+ϵ)+ϵ\text{ln}ϵ.}$

Lemma 3.2. Let us assume that (2.1) and (2.2) hold. Let (n_ϵ,p_ϵ) be a solution to (3.1)–(1.2). Then, for all T > 0, we have the uniform bounds in ϵ ∈ (0,ϵ₀),

$$\begin{array}{l}0\le n_{ϵ}\in L^{\infty }([0,T]);L^1\cap L^{\infty }({ℝ}^d));\\ 0\le p_{ϵ}\le P_M,0\le n_{ϵ}\le \frac{P_M}{P_M+ϵ}\le 1\end{array}.$$

More generally, we have the comparison principle: If n_ϵ, m_ϵ are respectively sub-solution and supersolution to (3.1), with initial data $n_{ϵ}^{ini},m_{ϵ}^{ini}.$ as in (2.2) and satisfying $n_{ϵ}^{ini}\le m_{ϵ}^{ini}.$ Then for all t > 0, n_ϵ(t) ≤ m_ϵ(t).

Finally, we have that (n_ϵ)_ϵ is uniformly bounded in L^∞([0,T],W^1,1 (ℝ^d)) and (p_ϵ)_ϵ is uniformly bounded in L¹([0,T ],W^1,1 (ℝ^d)).

Proof. Comparison principle.

Let n_ϵ be a subsolution and m_ϵ a supersolution of (3.1), we have

$${\partial }_t(n_{ϵ}-m_{ϵ})-\Delta (H(n_{ϵ})-H(m_{ϵ}))\le n_{ϵ}G(P(n_{ϵ}))-m_{ϵ}G(P(m_{ϵ})).$$

Notice that, since the function H is nondecreasing, the sign of n_ϵ − m_ϵ is the same as the sign of H(n_ϵ) − H(m_ϵ). Moreover,

$$\Delta f(y)=f^{″}(y)|\nabla y{|}^2+f^{\prime }(y)\Delta y,$$

so for y = H(n_ϵ) − H(m_ϵ) and f (y) = y₊ is the positive part, the so-called Kato inequality reads Δf (y) ≥ f′(y)Δy. Thus multiplying the latter equation by 1_{nϵ −mϵ>0}, we obtain

$${\partial }_t|n_{ϵ}-m_{ϵ}{|}_{+}-\Delta |H(n_{ϵ})-H(m_{ϵ}){|}_{+}\le |n_{ϵ}-m_{ϵ}|+G(P(n_{ϵ}))+m_{ϵ}(G(P(n_{ϵ}))-(G(P(m_{ϵ}))){\mathbf{\text{1}}}_{n_{ϵ}-m_{ϵ}>0}.$$

From assumption (2.1), we have that G is nonincreasing. Thus, since n ⟼ P (n) is increasing, we deduce that the last term of the right hand side is nonpositive. Since G is uniformly bounded we obtain

$${\partial }_t|n_{ϵ}-m_{ϵ}{|}_{+}-\Delta |H\left(n_{ϵ}\right)-H\left(m_{ϵ}\right){|}_{+}\le G_m|n_{ϵ}-m_{ϵ}{|}_{+}.$$

After an integration over ℝ^d,

$$\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}-m_{ϵ}{|}_{+}}dx\le G_m{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}-m_{ϵ}{|}_{+}}dx.$$

Then, integrating in time, we deduce

$${\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}-m_{ϵ}{|}_{+}}dx\le e^{G_{m}t}{\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}^{ini}-m_{ϵ}^{ini}{|}_{+}}dx.$$

Since we have $n_{ϵ}^{ini}\le m_{ϵ}^{ini},$ we deduce that for all t > 0, |n_ϵ − m_ϵ|₊(t) = 0.

L^∞ bounds.

We define $n_M=\frac{P_M}{ϵ+P_M},$ such that p_M = P (n_M), then applying the comparison principle with m_ϵ = n_M, we deduce, using also the assumption on the initial data (2.2) that for all 0 < ϵ ≤ ϵ₀, n_ϵ ≤ n_M. Moreover, since 0 is clearly a subsolution to (3.1), we also have by the comparison priniciple n_ϵ ≥ 0. Since n_M ≤ 1, we have 0 ≤ n_ϵ ≤ n_M ≤ 1 which implies

$$0\le p_{ϵ}\le P_M.$$

L¹ bound of n,p.

By nonnegativity, after a simple integration in space of equation (1.1), we deduce

$$\frac{d}{dt}\Vert n_{ϵ}{\Vert }_{L^1\left({ℝ}^d\right)}\le G_m\Vert n_{ϵ}{\Vert }_{L^1\left({ℝ}^d\right)},$$ (3.2)

where we use (2.1). Integrating in time give the L¹ bound,

$$\Vert n_{ϵ}{\Vert }_{L^1\left({ℝ}^d\right)}\le e^{G_{m}t}\Vert n_{ϵ}^{ini}{\Vert }_{L^1\left({ℝ}^d\right)}.$$

Then, using p_ϵ = n_ϵ(ϵ + p_ϵ) by (1.2), we get from the bound p_ϵ ≤ P_M, which has been proved above,

$$\Vert p_{ϵ}{\Vert }_{L^1\left(R^d\right)}\le \left(ϵ+P_M\right){\displaystyle {\int }_{{ℝ}^d}|n_{ϵ}|dx\le C{e}^{G_{m}t}}\Vert n_{ϵ}^{ini}{\Vert }_{L^1\left({ℝ}^d\right)}.$$

Estimates on the x derivative.

We derive equation (3.1) with respect to x_i for i = 1,…,d,

$${\partial }_t{\partial }_{x_i}{n}_{ϵ}-\Delta (H^{\prime }(n_{ϵ}){\partial }_{x_i}{n}_{ϵ})={\partial }_{x_i}{n}_{ϵ}G(p_{ϵ})+n_{ϵ}{G}^{\prime }(p_{ϵ}){\partial }_{x_i}{p}_{ϵ}.$$

Multiplying by sign(∂_xi n_ϵ), we get

$${\partial }_t|{\partial }_{x_i}{n}_{ϵ}|-\Delta ({\partial }_{x_i}H(n_{ϵ}))\text{sign}({\partial }_{x_i}{n}_{ϵ})=|{\partial }_{x_i}{n}_{ϵ}|G(p_{ϵ})+n_{ϵ}{G}^{\prime }(p_{ϵ}){\partial }_{x_i}{p}_{ϵ}\text{sign}({\partial }_{x_i}{n}_{ϵ}).$$

We can remark that sign(∂_xi n_ϵ) = sign(∂_xi H(n_ϵ)), so, by the same token as above, we have

$$\Delta ({\partial }_{x_i}H(n_{ϵ}))\text{sign}({\partial }_{x_i}{n}_{ϵ})\ge \Delta (|{\partial }_{x_i}H(n_{ϵ})|).$$

Moreover, sign(∂_xi n_ϵ) = sign(∂_xi p_ϵ), thus ∂_xi p_ϵsign(∂_xi n_ϵ) = |∂_xi p_ϵ|. By assumption (2.1), we know that

$$G^{\prime }\left(p_{ϵ}\right)\le -\gamma <0,$$

we deduce

$${\partial }_t|{\partial }_{x_i}{n}_{ϵ}|-\Delta (|{\partial }_{x_i}H(n_{ϵ})|)\le |{\partial }_{x_i}{n}_{ϵ}|G_m-\gamma n_{ϵ}|{\partial }_{x_i}{p}_{ϵ}|.$$

After an integration in time and space,

$$\Vert {\partial }_{x_i}{n}_{ϵ}{\Vert }_{L^1\left({ℝ}^d\right)}+\gamma {\displaystyle {\int }_0^t{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}|{\partial }_{x_i}{p}_{ϵ}|dxds\le e^{G_{m}t}}}\Vert {\partial }_{x_i}{n}_{ϵ}^{ini}{\Vert }_{L^1\left({ℝ}^d\right)}.$$ (3.3)

This latter inequality provides us with a uniform bound on the space derivative of n_ϵ in L¹. Then

$$\Vert {\partial }_{x_i}{p}_{ϵ}{\Vert }_{L^1\left({ℝ}^d\right)}={\displaystyle {\int }_{{ℝ}^d}|{\partial }_{x_i}{p}_{ϵ}|dx}={\displaystyle {\int }_{{ℝ}^d}\frac{ϵ}{{\left(1-n_{ϵ}\right)}^2}|{\partial }_{x_i}{n}_{ϵ}|dx.}$$

We split the integral in two: Either n_ϵ ≤ 1/2 and then $\frac{ϵ}{{\left(1-n_{ϵ}\right)}^2}\le C;$ or n_ϵ > 1/2.

$$\begin{array}{l}{\Vert {\partial }_{x_i}{p}_{ϵ}\Vert }_{L^1\left({\text{ℝ}}^d\right)}\le C{\displaystyle {\int }_{n_{ϵ}\le 1/2}|{\partial }_{x_i}{n}_{ϵ}|dx+{\displaystyle {\int }_{n_{ϵ}>1/2}|{\partial }_{x_i}{p}_{ϵ}|dx}}\\ \le C{\displaystyle {\int }_{n_{ϵ}\le 1/2}|{\partial }_{x_i}{n}_{ϵ}|dx+2{\displaystyle {\int }_{n_{ϵ}>1/2}\frac{1}{2}|{\partial }_{x_i}{p}_{ϵ}|dx}}\\ \le C{e}^{G_{m}t}{\displaystyle {\int }_{n_{ϵ}\le 1/2}|{\partial }_{x_i}{n}_{ϵ}^{ini}|dx+2{\displaystyle {\int }_{n_{ϵ}>1/2}{n}_{ϵ}|{\partial }_{x_i}{p}_{ϵ}|dx,}}\end{array}$$

where we have used the estimate (3.3) for the last inequality. Then, integrating in time, we deduce, using again the estimate (3.3)

$${\Vert {\partial }_{x_i}{p}_{ϵ}\Vert }_{L^1\left(Q_T\right)}\le C^{\prime }{e}^{G_{m}t}{\Vert {\partial }_{x_i}{n}_{ϵ}^{ini}\Vert }_{L^1\left({ℝ}^d\right)}.$$

It concludes the proof.

3.3. Compact support

The following Lemma proves that assuming that the initial data is compactly supported, then the pressure is compactly supported for any time with a control of the growth of the support.

Lemma 3.3 (Finite speed of propagation). Under the same assumptions as in Theorem 2.1, we have that supp p_ϵ ⊂ B(0,R(t)) with $R\left(t\right)\le 2\sqrt{C\left(T+t\right)},$ where B(0,R(t)) is the ball of center 0 and radius R(t).

Proof. Using the equation on p_ϵ (1.3),

$${\partial }_t{p}_{ϵ}-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\Delta p_{ϵ}-{|\nabla p_{ϵ}|}^2=(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ})\le G_m(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ}).$$

Let us introduce for C > 0,

$$\tilde{p}\left(t,x\right)={\left(C+\frac{{\left|x\right|}^2}{4\left(\theta +t\right)}\right)}_{+},$$

with $\theta =\frac{d}{4G_m}.$ Then p̃ is compactly supported in B(0,R_θ (t)) with $R_{\theta }(t)=2\sqrt{C(\theta +t)}.$ We have

$$\begin{array}{cc}{\partial }_t\tilde{p}=\frac{|x{|}^2}{4{(\theta +t)}^2}{1}_{|x|\le R_{\theta }(t)},& |\nabla \tilde{p}{|}^2\end{array}=\frac{|x{|}^2}{4{(\theta +t)}^2}{1}_{|x|\le R_{\theta }(t),}$$

and

$$\Delta \tilde{p}=-\frac{d}{(\theta +t)},\text{for}|x|<R_{\theta }(t).$$

Then, for all t ∈ [0,θ],

$${\partial }_t\tilde{p}-(\frac{{\tilde{p}}^2}{ϵ}+\tilde{p})\Delta \tilde{p}-|\nabla \tilde{p}{|}^2-G_m(\frac{{\tilde{p}}^2}{ϵ}+\tilde{p})=(\frac{{\tilde{p}}^2}{ϵ}+\tilde{p})(\frac{d}{(\theta +t)}-G_m)\ge 0.$$ (3.4)

In other words, p̃ is a supersolution for the equation for the pressure. Let us show that it implies that p ≤ p̃. We define $\tilde{n}=\frac{\tilde{p}}{ϵ+\tilde{p}}=N(\tilde{p}).$ We know that

$$N\prime (\tilde{p})=\frac{ϵ}{{(ϵ+\tilde{p})}^2}>0.$$

Then, on the one hand, multiplying (3.4) with by N′(p̃) we get

$${\partial }_t\tilde{n}-\nabla .(\tilde{n}\nabla \tilde{p})-G_m\tilde{n}\ge 0.$$

On the other hand, from (1.1),

$${\partial }_t{n}_{ϵ}-\nabla .(n_{ϵ}\nabla p_{ϵ})\le G_m{n}_{ϵ}.$$

By the comparison principle (see Lemma 3.2), we have

$$n_{ϵ}^{ini}\le {\tilde{n}}^{ini}\Rightarrow n_{ϵ}\le \tilde{n}.$$

Thus, for all t ∈ [0,θ],

$$p_{ϵ}^{ini}\le \tilde{p}(t=0)\Rightarrow p_{ϵ}\le \tilde{p}.$$

and p_ϵ(t) is compactly supported in B(0,R_θ(t)) provided we choose C large enough such that $p_{ϵ}^{ini}(x)\le \tilde{p}(t=0,x),$ which can be done thanks to our assumption on the initial data (2.2).

Since p_ϵ is uniformly bounded in L^∞, we may iterate the process on [θ,2θ]. After several iterations, we reach the time T and prove the result on [0,T ].

3.4. L² estimate for ∇_p

In the following Lemma, we state a uniform L² estimate on the gradient of the pressure.

Lemma 3.4 (L² estimate for ∇p). Under the same assumptions as in Theorem 2.1, we have a uniform bound on ∇p_ϵ in L²(Q_T).

Proof. For a given function ψ we have, multiplying (1.1) by ψ(n_ϵ),

$${\partial }_t{n}_{ϵ}\psi (n_{ϵ})-\nabla (n_{ϵ}\nabla p_{ϵ})\psi (n_{ϵ})=n_{ϵ}G(p_{ϵ})\psi (n_{ϵ}).$$

Let Ψ be an antiderivative of ψ, we have thanks to an integration by parts

$$\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}\Psi (n_{ϵ})dx+{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}}\nabla n_{ϵ}\cdot \nabla p_{ϵ}\psi \prime (n_{ϵ})dx={\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}G(p_{ϵ})\psi (n_{ϵ})dx.}}$$

We choose ψ such as n_ϵ∇n_ϵ · ∇p_ϵψ′(n_ϵ) = |∇p_ϵ|², i.e. n_ϵψ′(n_ϵ) = p′(n_ϵ). After straight-forward computations, we find $\psi (n)=ϵ(\text{ln}(n)-\text{ln}(1-n)+\frac{1}{1-n})$ and Ψ(n) = ϵn(ln(n) − ln(1 − n)). It gives

$$\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}}\text{ln}\left(\frac{n_{ϵ}}{1-n_{ϵ}}\right)dx+{\displaystyle {\int }_{{ℝ}^d}|\nabla p_{ϵ}{|}^{2}dx\le G_m{\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}|\text{ln}(n_{ϵ})-\text{ln}(1-n_{ϵ})+\frac{1}{1-n_{ϵ}}|dx.}}$$

We integrate in time, using also the expression of p_ϵ in (1.2),

$$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}\text{ln}\left(\frac{p_{ϵ}}{ϵ}\right)dx-{\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}^{ini}\text{ln}\left(\frac{n_{ϵ}^{ini}}{1-n_{ϵ}^{ini}}\right)dx+{\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^d}|\nabla p_{ϵ}{|}^{2}dxdt}}}}\\ \le G_m{\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^d}\left(ϵn_{ϵ}\left|\text{ln}\left(\frac{p_{ϵ}}{ϵ}\right)\right|+p_{ϵ}\right)dx.}}\end{array}$$

Then, to have a bound on the L²-norm of ∇p_ϵ, it suffices to prove a uniform control on ${\int }_{{ℝ}^d}ϵn_{ϵ}|\text{ln}(\frac{p_{ϵ}}{ϵ})|dx.$ We have

$${\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}|\text{ln}(\frac{p_{ϵ}}{ϵ})|dx\le {\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}|\text{ln}{p}_{ϵ}|dx+ϵ\text{ln}(ϵ){\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}dx.}}}$$

The second term of the right hand side is small when ϵ is small thanks to the L¹ bound on n_ϵ, thus it is uniformly bounded. Using the expression of p_ϵ in (1.2), we get

$${\displaystyle {\int }_{{ℝ}^d}ϵn_{ϵ}|\text{ln}(\frac{p_{ϵ}}{ϵ})|dx\le {\displaystyle {\int }_{{ℝ}^d}(1-n_{ϵ})p_{ϵ}|\text{ln}{p}_{ϵ}|dx+C.}}$$

Then, since 0 ≤ p_ϵ ≤ P_M and since x ↦ x| ln x | is uniformly bounded on [0,P_M], we get

$${\displaystyle {\int }_{{ℝ}^d}(1-n_{ϵ})p_{ϵ}|\text{ln}(p_{ϵ})|dx\le C{\displaystyle {\int }_{{ℝ}^d}{1}_{p_{ϵ}>0}dx.}}$$

We conclude thanks to Lemma 3.3, which provides a uniform control on the support of p_ϵ.

4. Regularizing effect and time compactness

As already noticed in [23], regularizing effects, similar to the ones observed for the heat equation [1, 10], allow to deduce estimates on the time derivatives.

Lemma 4.1. Under the assumptions (2.1) and (2.2), the weak solution (ρ_k,p_k) satisfies the estimates

$$\begin{array}{cc}{\partial }_t{p}_{ϵ}\ge -\frac{\kappa p_{ϵ}}{t},& {\partial }_t{n}_{ϵ}\ge -\frac{\kappa n_{ϵ}}{t},\end{array}$$

for a large enough (independent of ϵ) constant κ.

Proof. Let us denote w_ϵ = Δp_ϵ + G(p_ϵ), the equation on the pressure (1.3) reads

$${\partial }_t{p}_{ϵ}=\left(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ}\right)w_{ϵ}+|\nabla p_{ϵ}{|}^2.$$ (4.1)

The proof is divided into several steps. We first find a lower bound for w_ϵ by using the comparison principle. Then we deduce estimates on the density and on the pressure.

1st step. Thanks to (4.1), we deduce an equation satisfied by w_ϵ. On the one hand, by multiplying (4.1) by G′(p_ϵ), we deduce, since G is decreasing from (2.1)

$${\partial }_{t}G(p_{ϵ})\ge G\prime (p_{ϵ})(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})w_{ϵ}+2\nabla G(p_{ϵ})\cdot \nabla p_{ϵ}.$$ (4.2)

On the other hand, we have

$$\begin{array}{l}{\partial }_t\Delta p_{ϵ}=\Delta w_{ϵ}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})+2\nabla (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\cdot \nabla w_{ϵ}+w_{ϵ}\Delta (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\\ +2\nabla p_{ϵ}\cdot \nabla (\Delta p_{ϵ})+2{\displaystyle \sum _{i,j=1}^d{({\partial }_{x_i{x}_j}{p}_{ϵ})}^2}\\ \ge \Delta w_{ϵ}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})+2\nabla (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\cdot \nabla w_{ϵ}+w_{ϵ}\Delta (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\\ +2\nabla p_{ϵ}\cdot \nabla (\Delta p_{ϵ})+\frac{2}{d}({\Delta p}_{ϵ}{)}^2.\end{array}$$

Thus, with (4.2), we deduce that w_ϵ = Δp_ϵ + G(p_ϵ) satisfies

$$\begin{array}{l}{\partial }_t{w}_{ϵ}\ge \Delta w_{ϵ}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})+2\nabla (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\cdot \nabla w_{ϵ}+w_{ϵ}(\Delta p_{ϵ}(\frac{2p_{ϵ}}{ϵ}+1)+\frac{2}{ϵ}|\nabla p_{ϵ}{|}^2\\ +(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G\prime (p_{ϵ}))+2\nabla p_{ϵ}\cdot \nabla w_{ϵ}+\frac{2}{d}({\Delta p}_{ϵ}{)}^2.\end{array}$$

By definition of w_ϵ, we have ${(\Delta p_{ϵ})}^2\ge w_{ϵ}^2-2G(p_{ϵ})w_{ϵ}.$ Thus we deduce that

$${\partial }_t{w}_{ϵ}\ge ℱ(w_{ϵ}),$$ (4.3)

where we have used the notation

$$\begin{array}{l}ℱ(w):=\Delta w(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})+2\nabla (\frac{p_{ϵ}^2}{ϵ}+2p_{ϵ})\cdot \nabla w+\frac{2}{ϵ}|\nabla p_{ϵ}{|}^{2}w+w^2(\frac{2p_{ϵ}}{ϵ}+1+\frac{2}{d})\\ -w\left(G(p_{ϵ})(\frac{2p_{ϵ}}{ϵ}+1+\frac{4}{d})-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G\prime (p_{ϵ})\right).\end{array}$$ (4.4)

Following an idea of [10] which has been generalized in [23], we introduce the function

$$W(t,x)=-\frac{h(p_{ϵ}(t,x))}{t},$$ (4.5)

where the function h will be defined later such that W is a subsolution for (4.3). We compute

$$\begin{array}{l}{\partial }_{t}W=\frac{W^2}{h(p_{ϵ})}-\frac{h\prime (p_{ϵ})}{t}{\partial }_t{p}_{ϵ},\\ \begin{array}{cc}\nabla W=-\frac{h\prime (p_{ϵ})}{t}\nabla p_{ϵ},& \Delta W=-\frac{h\prime (p_{ϵ})}{t}\Delta p_{ϵ}-\frac{h″(p_{ϵ})}{t}|\nabla p_{ϵ}{|}^2.\end{array}\end{array}$$

Using again equation (4.1), we have

$$\begin{array}{l}{\partial }_{t}W=\frac{W^2}{h(p_{ϵ})}-\frac{h\prime (p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\Delta p_{ϵ}-\frac{h\prime (p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ})-\frac{h\prime (p_{ϵ})}{t}|\nabla p_{ϵ}{|}^2\\ =\frac{W^2}{h(p_{ϵ})}+(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})\Delta W+\frac{h″(p_{ϵ})}{t}|\nabla p_{ϵ}{|}^2(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})-\frac{h\prime (p_{ϵ})}{t}|\nabla p_{ϵ}{|}^2\\ -\frac{h\prime (p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ}).\end{array}$$ (4.6)

By definition of ℱ(W) in (4.4), we deduce with (4.6),

$$\begin{array}{l}{\partial }_{t}W=ℱ(W)+4(\frac{p_{ϵ}}{ϵ}+1){\left|\nabla p_{ϵ}\right|}^2\frac{h^{\prime }(p_{ϵ})}{t}+\frac{2}{ϵ}\frac{h(p_{ϵ})}{t}{\left|\nabla p_{ϵ}\right|}^2\\ +W^2\left(\frac{1}{h(p_{ϵ})}-\frac{2p_{ϵ}}{ϵ}-1-\frac{2}{d}\right)+\frac{h^{″}(p_{ϵ})}{t}{\left|\nabla p_{ϵ}\right|}^2(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})-\frac{h^{\prime }(p_{ϵ})}{t}{\left|\nabla p_{ϵ}\right|}^2\\ -\frac{h^{\prime }(p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ})+W\left(G(p_{ϵ})(\frac{2p_{ϵ}}{ϵ}+1+\frac{4}{d})-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G^{\prime }(p_{ϵ})\right).\end{array}$$

We may rearrange it into

$$\begin{array}{l}{\partial }_{t}W=ℱ(W)+W^2\left(\frac{1}{h(p_{ϵ})}-\frac{2p_{ϵ}}{ϵ}-1-\frac{2}{d}\right)+\frac{{\left|\nabla p_{ϵ}\right|}^2}{t}\left({(h(p_{ϵ})(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ}))}^{″}+h^{\prime }(p_{ϵ})\right)\\ -\frac{h^{\prime }(p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G(p_{ϵ})+W\left(G(p_{ϵ})(\frac{2p_{ϵ}}{ϵ}+1+\frac{4}{d})-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G^{\prime }(p_{ϵ})\right).\end{array}$$ (4.7)

Let us choose

$$h(p)=\frac{\kappa ϵ}{p+ϵ},$$ (4.8)

where κ > 0 is chosen large enough (independent of ϵ) such that

$$\frac{1}{h(p_{ϵ})}=\frac{p_{ϵ}+ϵ}{\kappa ϵ}\le \frac{2p_{ϵ}}{ϵ}+1+\frac{2}{d}.$$

Thanks to this choice, we have

$$(h(p_{ϵ})(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ}))″+h\prime (p_{ϵ})=-\frac{\kappa ϵ}{{(p_{ϵ}+ϵ)}^2}\le 0,$$

and

$$-\frac{h^{\prime }(p_{ϵ})}{t}(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})=W\frac{p_{ϵ}}{ϵ}.$$

Finally, we obtain from (4.7)

$${\partial }_{t}W\le ℱ(W)+W\left(G(p_{ϵ})(\frac{p_{ϵ}}{ϵ}+1+\frac{4}{d})-(\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})G^{\prime }(p_{ϵ})\right)\le ℱ(W),$$

where we use the fact that by definition (4.5) we have W ≤ 0 (recalling also that G is decreasing by assumption (2.1)).

Thus, by the sub- and super-solution technique, we deduce, using also (4.3) that

$$w_{ϵ}\ge W=-\frac{\kappa ϵ}{t(p_{ϵ}+ϵ)}.$$ (4.9)

2nd step. Using again equation (4.1), we get from (4.9)

$${\partial }_t{p}_{ϵ}\ge (\frac{p_{ϵ}^2}{ϵ}+p_{ϵ})W=-\frac{\kappa p_{ϵ}}{t},$$

which is the first inequality of Lemma 4.1. Finally, by definition (1.2), we have also $n_{ϵ}=\frac{p_{ϵ}}{p_{ϵ}+ϵ}.$ Thus

$${\partial }_t{n}_{ϵ}=\frac{ϵ}{{(p_{ϵ}+ϵ)}^2}{\partial }_t{p}_{ϵ}\ge -\frac{\kappa ϵp_{ϵ}}{t{(p_{ϵ}+ϵ)}^2}=-\frac{\kappa n_{ϵ}(1-n_{ϵ})}{t},$$

where we use the definition (1.2) for the last identity. We conclude easily the proof.

Thanks to this latter Lemma, we may deduce uniform estimates on the time derivative of n_ϵ and p_ϵ.

Lemma 4.2. For any τ > 0, we have that ∂_tn_ϵ is uniformly bounded in L^∞([τ,T];L¹(ℝ^d)) and ∂_tp_ϵ is uniformly bounded in L¹([τ,T] × ℝ^d).

Proof. We use the equality |∂_tn_ϵ| = ∂_tn_ϵ + 2|∂_tn_ϵ|₋, where we recall that |·|₋ denotes the negative part. Thus

$$\begin{array}{l}{\Vert {\partial }_t{n}_{ϵ}\Vert }_{L^1({ℝ}^d)}=\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}dx}+2{\displaystyle {\int }_{{ℝ}^d}\left|{\partial }_t{n}_{ϵ}\right|-dx}\\ \le \left(G_m+\frac{2\kappa }{t}\right){\Vert n_{ϵ}\Vert }_{L^1({ℝ}^d)},\end{array}$$

where we have used equation (3.2) to bound the first term and Lemma 4.1 for the second term. By the same token, we have

$$\begin{array}{l}{\Vert {\partial }_t{p}_{ϵ}\Vert }_{L^1(\left[\tau ,T\right]\times {ℝ}^d)}={\displaystyle {\int }_{\tau }^T\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}{p}_{ϵ}dx}+2{\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{{ℝ}^d}{|{\partial }_t{p}_{ϵ}|}_{-}dx}}}\\ \le {\Vert p_{ϵ}(T)\Vert }_{L^1({ℝ}^d)}+{\Vert p_{ϵ}\Vert }_{L^{\infty }(\left[\tau ,T\right];L^1({ℝ}^d))}2\kappa \text{ln}(T/\tau ).\end{array}$$

We conclude the proof thanks to the estimates on n_ϵ and p_ϵ in L¹∩L^∞ obtained in Lemma 3.2.

5. Convergence

This section is devoted to the proof of Theorem 2.1 apart from the complementary relation (2.7) which is postponed to the next section.

Since the sequences (n_ϵ)_ϵ and (p_ϵ)_ϵ are bounded in $W_{loc}^{1,1}(Q_T),$ due to Lemma 3.2 and 4.2, we may apply the Helly theorem and recover strong convergence in $L_{loc}^1(Q_T),$ up to an extraction. If we want to extend this local convergence to a global convergence in L¹(Q_T) we need to prove that we can control the mass in an initial strip and in the tail. Indeed, let ϵ,ϵ′ > 0, R > 0, τ > 0

$$\begin{array}{l}{\Vert n_{ϵ}-n_{{ϵ}^{\prime }}\Vert }_{L^1(Q_T)}={\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^d}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dx}dt}\\ \le {\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{B(0,R)}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dx}dt}\\ +{\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{{ℝ}^d\backslash B(0,R)}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dx}dt}\\ +{\displaystyle {\int }_0^{\tau }{\displaystyle {\int }_{{ℝ}^d}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dx}dt.}\end{array}$$

Since we have strong convergence of n_ϵ in $L_{loc}^1(Q_T),$

$${\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{B(0,R)}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dx}dt}\underset{ϵ\to 0}{\to }0.$$

Then we have to control the two other terms in the right hand side.

The control of the initial strip comes from the L¹ estimate of n,

$${\displaystyle {\int }_0^{\tau }{\displaystyle {\int }_{{ℝ}^d}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dxdt}\le {\displaystyle {\int }_0^{\tau }\left({\Vert n_{ϵ}(t,x)\Vert }_{L^1({ℝ}^d)}+{\Vert n_{{ϵ}^{\prime }}(t,x)\Vert }_{L^1({ℝ}^d)}\right)}}\underset{\tau \to 0}{dt\to 0}$$

For the control of the tail we consider ϕ ∈ C^∞(ℝ) such that 0 ≤ ϕ ≤ 1, ϕ(x) = 0 for |x| < R − 1 and ϕ(x) = 1 for |x| > R. We define ϕ_R(x) = ϕ(x/R). Then

$$\begin{array}{l}{\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{{ℝ}^d\backslash B(0,R)}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|dxdt}\le {\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{{ℝ}^d\backslash B(0,R)}\left|n_{ϵ}(t,x)-n_{{ϵ}^{\prime }}(t,x)\right|}}}{\varphi }_{R}dxdt\\ \le {\displaystyle {\int }_{\tau }^T{\displaystyle {\int }_{{ℝ}^d\backslash B(0,R)}(n_{ϵ}(t,x)+n_{{ϵ}^{\prime }}(t,x)){\varphi }_{R}dxdt,}}\end{array}$$

where the notation C stand for a generic nonnegative constant. Moreover, using equation (3.1), we deduce

$$\begin{array}{l}\frac{d}{dt}{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}{\varphi }_{R}dx}={\displaystyle {\int }_{{ℝ}^d}H(n_{ϵ})\Delta {\varphi }_{R}dx+{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}G(p_{ϵ}){\varphi }_{R}dx}}\\ \le C{R}^{-2}{\Vert \Delta \varphi \Vert }_{L^{\infty }}+G_m{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}}{\varphi }_{R}dx.\end{array}$$

Then, integrating on [0,T], we get

$$\begin{array}{l}0\le {\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}{\varphi }_{R}dx\le e^{G_{m}T}\left({\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}^{ini}{\varphi }_R+C{R}^{-2}T}\right)}\\ \le e^{G_{m}T}\left({\Vert n_{ϵ}^{ini}-n^{ini}\Vert }_{L^1({ℝ}^d)}+{\displaystyle {\int }_{{ℝ}^d}{n}^{ini}{\varphi }_{R}dx}+C{R}^{-2}T\right).\end{array}$$

By assumption (2.2), since the initial data is uniformly compactly supported, we deduce that the right hand side tends to 0 as R goes to +∞ and ϵ goes to 0. Then (n_ϵ)_ϵ is a Cauchy sequence in L¹(Q_T). It implies its convergence in L¹(Q_T). The convergence of the pressure follows from the same kind of computation. The only difference is for the control of the tail and which is directly given by the estimate

$$0\le {\displaystyle {\int }_{{ℝ}^d}{p}_{ϵ}{\varphi }_{R}dx}\le (ϵ+P_M){\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}{\varphi }_{R}dx}.$$

Therefore, we can extract subsequences and pass to the limit in the equation

$$(1-n_{ϵ})p_{ϵ}=ϵn_{ϵ},$$

which implies

$$(1-n_0)p_0=0.$$

This is the relation (2.6). We can also pass to the limit in the uniform estimate of Lemma 3.2 which provides (2.3) and n₀,p₀ ∈ BV (Q_T).

Limit model. We first recall that from (3.1), we have

$${\partial }_t{n}_{ϵ}-\Delta (p_{ϵ}-ϵ\text{ln}(p_{ϵ}+ϵ))=n_{ϵ}G(p_{ϵ}).$$

We get,

$$ϵ\text{ln}ϵ\le ϵ\text{ln}(p_{ϵ}+ϵ)\le ϵ\text{ln}(P_M+ϵ).$$

Thus, the term in the Laplacien converges strongly to p₀ as ϵ goes to 0. Then, thanks to the strong convergence of n_ϵ and p_ϵ, we deduce that in the sense of distribution (n₀,p₀) satisfies (2.4). Moreover, due to the uniform estimate on ∇p in L²(Q_T) of Lemma 3.4, we can show, by passing into the limit in a product of a weak-strong convergence, that in the sense of distribution (n₀,p₀) satisfies (2.5).

Time continuity. Let us define 0 < t₁ < t₂ ≤ T, η > 0. For a given R > 0, we consider a smooth function ζ_R on ℝ^d such that 0 ≤ ζ_R ≤ 1, ζ_R(x) = 1 for |x| < R − 1 and ζ_R(x) = 0 for |x| > R. We have

$${\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|dx=}{\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|{\zeta }_{R}dx+}{\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|(1-{\zeta }_R)dx.}$$

We have

$${\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|(1-{\zeta }_R)dx\le {\displaystyle {\int }_{{ℝ}^d}{n}_0(t_2)}(1-{\zeta }_R)dx+{\displaystyle {\int }_{{ℝ}^d}{n}_0(t_1)}(1-{\zeta }_R)dx}$$

with 1 − ζ_R a function which is zero on B(0, R−1). Thus, as for the control of the tail, for R large enough, we have, uniformly for 0 < t₁ < t₂ ≤ T,

$${\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|(1-{\zeta }_R)dx\le \eta .}$$

In addition, we know from Lemma 4.1 (and the L^∞ bound on n₀) that ${\partial }_t{n}_0\ge -\frac{C}{t},$ so ∂_t(n₀ + Cln(t)) ≥ 0. Then, since t₁ < t₂,

$$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|{\zeta }_R)dx\le {\displaystyle {\int }_{{ℝ}^d}(n_0(t_2)+}C\text{ln}(t_2)-(n_0(t_1)+C\text{ln}(t_1)){\zeta }_{R}dx}\\ +{\displaystyle {\int }_{{ℝ}^d}C(\text{ln}(t_2)-\text{ln}(t_1)){\zeta }_{R}dx}\\ \le {\displaystyle {\int }_{t_1}^{t_2}{\displaystyle {\int }_{{ℝ}^d}{\partial }_t(n_0+C\text{ln}(t)){\zeta }_R)dxdt+{\displaystyle {\int }_{{ℝ}^d}C\text{ln}(t_2)-\text{ln}(t_1)){\zeta }_{R}dx.}}}\end{array}$$

Then, using equation (2.4) and an integration by parts, we obtain

$$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|{\zeta }_R)dx\le }{\displaystyle {\int }_{t_1}^{t_2}{\displaystyle {\int }_{{ℝ}^d}(p_0\Delta {\zeta }_R+n_{0}G(p_0){\zeta }_R)dxdt}}\\ +2{\displaystyle {\int }_{{ℝ}^d}C(\text{ln}(t_2)-\text{ln}(t_1))}{\zeta }_{R}dx\\ \le C(t_2-t_1)(||\Delta {\zeta }_R|{|}_{\infty }+1)+2C(\text{ln}(t_2)-\text{ln}(t_1)){\displaystyle {\int }_{{ℝ}^d}{\zeta }_{R}dx.}\end{array}$$

Then we can choose (t₁,t₂) close enough such that

$${\displaystyle {\int }_{{ℝ}^d}|n_0(t_2)-n_0(t_1)|{\zeta }_R)dx\le }\eta .$$

We conclude that n₀ ∈ C((0,T),L¹(ℝ^d)).

Initial trace For any test function 0 ≤ ζ(x) ≤ 1, we have from (3.1),

$$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}}(t)\zeta dx-{\displaystyle {\int }_{{ℝ}^d}{n}_{ϵ}^{ini}\zeta dx={\displaystyle {\int }_0^t{\displaystyle {\int }_{{ℝ}^d}(\Delta H(n_{ϵ})+n_{ϵ}G(p_{ϵ}))\zeta dxds}}}\\ ={\displaystyle {\int }_0^t{\displaystyle {\int }_{{ℝ}^d}(H(n_{ϵ})\Delta \zeta +n_{ϵ}G(p_{ϵ})\zeta )dxds.}}\end{array}$$

Letting ϵ going to 0, we obtain with (2.2),

$${\displaystyle {\int }_{{ℝ}^d}{n}_0}(t)\zeta dx-{\displaystyle {\int }_{{ℝ}^d}{n}_0^{ini}\zeta dx={\displaystyle {\int }_0^t{\displaystyle {\int }_{{ℝ}^d}(p_0\Delta \zeta +n_{0}G(p_0)\zeta )}}}dxds.$$

Letting t → 0 we can conclude that $n_0(0)=n_0^{ini}.$

6. Complementary relation

In this section we prove the complementary relation

$$p_0^2(\Delta p_0+G(p_0))=0.$$

In the weak sense, this identity reads, for any test function ϕ,

$${\displaystyle {\iint }_{Q_T}(-2\varphi p_0|\nabla p_0{|}^2-p_0^2\nabla p_0\cdot \nabla \varphi +\varphi p_0^{2}G(p_0))dxdt=0}.$$ (6.1)

The proof is divided into two steps.

1st step. In this first step we prove the inequality ≥ 0 in (6.1). We start with the pressure equation (1.3) that we multiply by ϵ

$$ϵ{\partial }_t{p}_{ϵ}-p_{ϵ}(ϵ+p_{ϵ})\Delta p_{ϵ}-ϵ|\nabla p_{ϵ}{|}^2=p_{ϵ}(ϵ+p_{ϵ})G(p_{ϵ}).$$

We multiply by a test function $\varphi \in 𝒟\left(\right(0,T)\times {ℝ}^d)$ and integrate,

$$\begin{array}{l}{\displaystyle {\iint }_{Q_T}{p}_{ϵ}^2\varphi (\Delta p_{ϵ}+G(p_{ϵ}))}dxdt=ϵ{\displaystyle {\iint }_{Q_T}\varphi ({\partial }_t{p}_{ϵ}-|\nabla P_{ϵ}{|}^2-P_{ϵ}(\Delta p_{ϵ}+G(p_{ϵ}))dxdt}\\ =ϵ{\displaystyle {\iint }_{Q_T}(\varphi {\partial }_t{p}_{ϵ}+p_{ϵ}\nabla P_{ϵ}\cdot \nabla \varphi -\varphi p_{ϵ}G(p_{ϵ}))}dxdt,\end{array}$$

where we use an integration by parts for the last identity. From the estimates in Lemma 3.2, we have

$$\begin{array}{l}\left|ϵ{\displaystyle {\iint }_{Q_T}(\varphi {\partial }_t{p}_{ϵ}+p_{ϵ}\nabla P_{ϵ}\cdot \nabla \varphi -\varphi p_{ϵ}G(p_{ϵ}))dxdt}\right|\\ \le ϵ(\Vert \varphi {\Vert }_{L^{\infty }}\Vert {\partial }_t{p}_{ϵ}{\Vert }_{L^1(Q_T)}+\Vert \nabla \varphi {\Vert }_{L^{\infty }}{P}_M\Vert {\nabla }_{p_{ϵ}}{\Vert }_{L^1(Q_T)}+\Vert \varphi {\Vert }_{L^{\infty }}{G}_m\Vert p_{ϵ}{\Vert }_{L^1(Q_T)})\underset{ϵ\to 0}{\to }0\end{array}.$$

We deduce that for any test function $\varphi \in 𝒟\left(\right(0,T)\times {ℝ}^d)$,

$${\displaystyle {\iint }_{Q_T}(-2\varphi p_{ϵ}|\nabla p_{ϵ}{|}^2-p_{ϵ}^2\nabla p_{ϵ}\nabla \varphi +\varphi p_{ϵ}^{2}G(p_{ϵ}))dxdt\underset{ϵ\to 0}{\to }}0.$$ (6.2)

Since we have strong convergence of (p_ϵ)_ϵ and weak convergence of (∇p_ϵ)_ϵ, we can pass into the limit in the last two term in (6.2),

$${\displaystyle {\iint }_{Q_T}(-p_{ϵ}^2\nabla p_{ϵ}\nabla \varphi +\varphi p_{ϵ}^{2}G(p_{ϵ}))}dxdt\underset{ϵ\to 0}{\to }{\displaystyle {\iint }_{Q_T}(-p_0^2\nabla p_0\nabla \varphi +\varphi p_0^{2}G(p_0))}dxdt.$$

Now we are looking for the limit of the first term in (6.2). We have $p_{ϵ}|\nabla p_{ϵ}{|}^2=\frac{4}{9}|\nabla p_{ϵ}^{3/2}{|}^2.$ By weak convergence of $\nabla p_{ϵ}^{3/2}=p_{ϵ}^{1/2}\nabla p_{ϵ}$ and with Jensen inequality (since x↦x² is convex),

$$\underset{ϵ\to 0}{\lim \inf }{\displaystyle {\iint }_{Q_T}\varphi p_{ϵ}}|\nabla p_{ϵ}{|}^{2}dxdt\le {\displaystyle {\iint }_{Q_T}\varphi }|\nabla p_0^{3/2}{|}^{2}dxdt.$$

Thus, we conclude from (6.2) that

$$0\le {\displaystyle {\iint }_{Q_T}(-2\varphi p_0|\nabla p_0{|}^2-p_0^2\nabla p_0\nabla \varphi +\varphi p_0^{2}G(p_0))}dxdt,$$

which is a first inequality for (6.1).

2nd step. Now we want to show the reverse inequality, i.e.

$$0\ge {\displaystyle {\iint }_{Q_T}(-2\varphi p_0|\nabla p_0{|}^2-p_0^2\nabla p_0\nabla \varphi +\varphi p_0^{2}G(p_0))}dxdt.$$

We know that

$${\partial }_t{n}_{ϵ}-\Delta q_{ϵ}=n_{ϵ}G(p_{ϵ}),$$

with q_ϵ = p_ϵ − ϵln(p_ϵ + ϵ). Thanks to the inequality ϵln(ϵ) ≤ ϵln(p_ϵ + ϵ) ≤ ϵln(p_M + ϵ), and the strong convergence p_ϵ → p₀, we know that q_ϵ → p₀ as ϵ → 0. Because

$$\Delta q_{ϵ}={\partial }_t{n}_{ϵ}-n_{ϵ}G(p_{ϵ}),$$

we deduce from Lemma 3.2 that ∇q_ϵ ∈ L^∞([0,T];L¹(ℝ^d)). It gives us compactness in space but not in time. Thus, following the idea of [22], we use a regularization process ’à la Steklov’.

Let introduce a time regularizing kernel ω_η ≥ 0 such that supp(ω_η) ⊂ ℝ₋. Then with the notations n_ϵ,η = ω_η*t n_ϵ, q_ϵ,η = ω_η*t q_ϵ, where the convolution holds only in the time variable,

$${\partial }_t{n}_{ϵ,\eta }-\Delta q_{ϵ,\eta }=(n_{ϵ}G(p_{ϵ}))\ast {\omega }_{\eta }$$ (6.3)

We denote U_ϵ = Δq_ϵ,η, then

$$\begin{array}{l}{U}_{ϵ}={\partial }_t{n}_{ϵ,\eta }-(n_{ϵ}G(p_{ϵ}))\ast {\omega }_{\eta }\\ =n_{ϵ}\ast {\partial }_t{\omega }_{\eta }-(n_{ϵ}G(p_{ϵ}))\ast {\omega }_{\eta }\end{array}$$

Since n_ϵ and n_ϵG(p_ϵ) are uniformly bounded in W^1,1(Q_T) from Lemma 3.2, (U_ϵ)_ϵ is bounded in W^1,1(Q_T) and we can extract a converging subsequence, still denoted (U_ϵ)_ϵ, converging towards U₀ in $L_{loc}^1({ℝ}^d)$ for η fixed. Moreover

$$U_0=\Delta p_0\ast {\omega }_{\eta }.$$

We multiply (6.3) by $P^{\prime }(n_{ϵ})=\frac{ϵ}{{(1-n_{ϵ})}^2}=\frac{1}{ϵ}{(p_{ϵ}+ϵ)}^2,$

$$\frac{ϵ}{{(1-n_{ϵ})}^2}{\partial }_t{n}_{ϵ,\eta }-\frac{1}{ϵ}{(p_{ϵ}+ϵ)}^2\Delta q_{ϵ,\eta }=\frac{1}{ϵ}{(p_{ϵ}+ϵ)}^2(n_{ϵ}G(p_{ϵ}))*{\omega }_{\eta }.$$

Then, passing to the limit ϵ→0, we obtain, thanks to the above remark

$$\frac{{ϵ}^2}{{(1-n_{ϵ})}^2}{\partial }_t{n}_{ϵ,\eta }\underset{ϵ\to 0}{\to }{p}_0^2\Delta p_0*{\omega }_{\eta }+p_0^2(n_{0}G(p_0))*{\omega }_{\eta }.$$

So we are left to prove that for any η > 0, we have

$$\underset{ϵ\to 0}{\text{lim}}\frac{{ϵ}^2}{{(1-n_{ϵ})}^2}{\partial }_t{n}_{ϵ,\eta }\le 0.$$

We compute for a fixed η > 0,

$$\begin{array}{l}\frac{{ϵ}^2}{{(1-n_{ϵ})}^2}{\partial }_t{n}_{ϵ,\eta }(t,x)={\displaystyle {\int }_{ℝ}\frac{{ϵ}^2}{{(1-n_{ϵ}(t,x))}^2}}{\partial }_t{n}_{ϵ}(s,x){\omega }_{\eta }(t-s,x)ds\\ ={\displaystyle {\int }_{ℝ}\frac{{ϵ}^2}{{(1-n_{ϵ}(s,x))}^2}}{\partial }_t{n}_{ϵ}(s,x){\omega }_{\eta }(t-s,x)ds\\ +{\displaystyle {\int }_{ℝ}(\frac{{ϵ}^2}{{(1-n_{ϵ}(t,x))}^2}-\frac{{ϵ}^2}{{(1-n_{ϵ}(s,x))}^2})({\partial }_t{n}_{ϵ}(s,x+\frac{C}{s}){\omega }_{\eta }(t-s,x)ds}\\ -C{\displaystyle {\int }_{ℝ}(\frac{{ϵ}^2}{{(1-n_{ϵ}(t,x))}^2}-\frac{{ϵ}^2}{{(1-n_{ϵ}(s,x))}^2})\frac{{\omega }_{\eta }(t-s,x)}{s}ds}\\ ={\text{I}}_{ϵ}+{\text{II}}_{ϵ}+{\text{III}}_{ϵ},\end{array}$$

where C is a constant such that ${\partial }_t{n}_{ϵ}(s,x)+\frac{C}{t}\ge 0.$

For the first term we have

$$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^d}|I_{ϵ}}|dxds\le ϵ{\displaystyle \int {\displaystyle {\int }_{Q_T}|{\partial }_t{p}_{ϵ}(s,x)|{\omega }_{\eta }(t-s,x)}dxds}\\ \le ϵ\Vert {\omega }_{\eta }{\Vert }_{L^{\infty }}\Vert {\partial }_t{p}_{ϵ}{\Vert }_{L_1(Q_T)}\le ϵC_{\eta }\\ \underset{ϵ\to 0}{\to }0.\end{array}$$

For the second term, we have

$$\frac{{ϵ}^2}{{(1-n_{ϵ}(t,x))}^2}={(p_{ϵ}+ϵ)}^2$$

and ${\partial }_t{(p_{ϵ}+ϵ)}^2=2(p_{ϵ}+ϵ){\partial }_t{p}_{ϵ}\ge -\frac{C^{\prime }}{t}.$ Let $0\le \xi \in {𝒞}_c^{\infty }(Q)$ and τ > 0 the smallest time in its support, we then have for t ≥ τ

$${\partial }_t{(p_{ϵ}+ϵ)}^2(t,x)\ge -\frac{C^{\prime }}{\tau }.$$

So integrating on (t,s) ⊂ (τ,+∞)

$$\frac{{ϵ}^2}{{(1-n_{ϵ}(t,x))}^2}-\frac{{ϵ}^2}{{(1-n_{ϵ}(s,x))}^2}\le \frac{C^{\prime }}{\tau }(s-t).$$

Then

$${\displaystyle \int {\displaystyle {\int }_Q\xi {\text{II}}_{ϵ}}}\le \frac{C^{\prime }}{\tau }\eta {\displaystyle \int {\displaystyle {\int }_Q{\displaystyle {\int }_{ℝ}({\partial }_t{n}_{ϵ}(s,x)+\frac{C}{\tau })}}}{\omega }_{\eta }(t-s,x)dsdxdt\le C_{\tau }^{\prime }\eta ,$$

where we use the bound on ∂_tn in Lemma 4.2.

For the third term, since s ≥ t > 0, for any test function ξ as above,

$$\begin{array}{l}{\displaystyle \int {\displaystyle {\int }_Q\xi {\text{III}}_{ϵ}}}=-C{\displaystyle \int {\displaystyle {\int }_Q\xi }}{\displaystyle {\int }_{ℝ}({(p_{ϵ}(t)+ϵ)}^2-{(p_{ϵ}(s)+ϵ)}^2)\frac{{\omega }_{\eta }(t-s,x)}{s}}ds\\ \underset{ϵ\to 0}{\to }-C{\displaystyle \int {\displaystyle {\int }_Q\xi }}{\displaystyle {\int }_{ℝ}(p_0{(t)}^2-p_0{(s)}^2)\frac{{\omega }_{\eta }(t-s,x)}{s}}dsdxdt\\ \underset{ϵ\to 0}{\to }-C{\displaystyle \int {\displaystyle {\int }_Q\xi }}\left[p_0^2(t){\displaystyle {\int }_{ℝ}\frac{{\omega }_{\eta }(t-s,x)}{s}ds-{\displaystyle {\int }_{ℝ}\frac{p_0{(s)}^2}{s}}{\omega }_{\eta }(t-s,x)ds}\right]dxdt\\ =\underset{\eta \to 0}{o}(1).\end{array}$$

So for all test function ξ as above, and all η > 0,

$${\displaystyle \int {\displaystyle {\int }_{Q_T}\xi (p_0^2\Delta p_0\ast {\omega }_{\eta }+p_0^2(n_{0}G(p_0))\ast {\omega }_{\eta })dxdt\le \underset{\eta \to 0}{o}(1).}}$$

Now it remain to pass to the limit η → 0 in the regularization process. Thanks to an integration by parts,

$$0\ge {\displaystyle \int {\displaystyle {\int }_{Q_T}(-2\xi p_0\nabla p_0\cdot \nabla p_0\ast {\omega }_{\eta }-p_0^2\nabla \xi \cdot \nabla p_0\ast {\omega }_{\eta }+\xi p_0^2(n_{0}G(p_0))\ast {\omega }_{\eta })dxdt.}}$$

From the L² estimate on ∇p₀ (Lemma 3.4) and the L¹∩L^∞ estimate on p₀ (Lemma 3.2), we deduce that we can pass to the limit η → 0 and get

$$0\ge {\displaystyle \int {\displaystyle {\int }_{Q_T}(-2\xi p_0|\nabla p_0{|}^2-p_0^2\nabla \xi \cdot \nabla p_0+\xi p_0^2{n}_{0}G(p_0))dxdt}.}$$

Finally, from (2.6), we have p₀n₀ = p₀. It concludes the proof.