Convergence rate for the incompressible limit of nonlinear diffusion–advection equations

<jats:p>The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cellpopulation models to free boundary problems of Hele–Shaw type. Although a vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.</jats:p>

section_content: We consider the nonlinear drift-diffusion equation @n @t r. nrp C nrV / D ng;

(1) posed on. 0; T / R d, d 2, where n describes a population density and p D p. n/ is the density-dependent pressure. The reaction term on the right-hand side represents the population growth rate g D g. t; x/, while V D V. t; x/ is a chemical concentration. The pressure is assumed to be a known increasing function of the density. We consider the following two representative examples:

and p " D P ". n/ WD " n 1 n

; " > 0:

We are concerned with calculating the rate at which solutions to (1) converge to the so-called incompressible (or stiff pressure) limit, as described below. More precisely we prove the following results. 

Theorem 1. 1 (Convergence rate in P H 1 ). Assume (A-L 1 data), (A-drift), and (A-reaction) (for d D 2) and (A-reaction 0 ) or (A-reaction 00 ) (for d 3). For d D 2 assume additionally (A-2D). Then, for all T > 0, there exists a unique function n 1 2 C. OE0; T /I L 1. R d // such that the sequence n (resp. n " ) converges, as ! 1 (resp. " ! 0), to n 1 strongly in L 1

Theorem 1. 3. Under the assumptions of Theorem 1. 1, there exists a function p 1 2 L 1. . 0; T / R d / such that, after extracting a subsequence, the sequence p converges to p 1 weakly in L 1. . 0; T / R d / and the relation

holds almost everywhere in. 0; T / R d. 

The above graph relation between the limit pressure and density is well known in the literature. In particular, when considering tumor growth models it implies that saturation holds in the region where there is a positive pressure, which is usually referred to as the region occupied by the tumor. Here we provide a new proof that does not require strong convergence of the density or the pressure. 

In fact, the limit n 1 satisfies (together with a limit pressure, p 1 ) a free-boundary-type problem, discussed briefly below, and the question of passing to this limiting problem has been vastly addressed in literature. Our contribution is to provide a new proof together with a convergence rate. 

Motivation and previous works. Models like (1) are well known and commonly employed in a variety of applications, for instance in bio-mathematical modeling of living tissue. In the case V D 0, g D 0, it is well known that if the pressure satisfies the power law (2), then (1) is actually the porous medium equation

whose well-understood properties (e. g. regularizing effects) facilitate the analysis notably. 

The other choice of the pressure, given by (3), is well known in kinetic theory of dense gases where the short-distance interactions between particles are strongly repulsive. In this spirit it has been used in models describing collective motion or congested traffic flow; see [2, 3, 12, 19, 24]. Despite having a singularity when the population density reaches its maximum value (here standardized to 1), this choice of pressure gives rise to a tissue growth model with similar properties -indeed, the crucial a priori estimates are the same and the limiting free-boundary-type problem is almost identical. A difference is that the singularity in the pressure prevents the cell densities ever rising above the maximum value 1. Taking advantage of these similarities, we shall henceforth index the solution of (1) by, n D n, and consider the singular limit ! 1. Each of the assumptions and properties we discuss below has its natural "-analogue by putting " D 1=. 

Let us recall that the study of the incompressible limit has a long history and it has been investigated for many different models related to (1). The first result on the limit ! 1 was obtained for the classical porous medium equation (6). The most interesting difference from the case with a non-trivial reaction term is that the free boundary problem arising in the limit turns out to be stationary. In fact, as proven in [5] the limit density n 1 is independent of time. This result can be intuitively explained by noticing that the degenerate diffusivity of (1), namely n 1, converges to 0 if n < 1, while it tends to infinity in the regions where n > 1. Therefore, while there is no motion in the regions where the density is below 1, where the solution lies above this level it tends to collapse instantaneously; cf. [16]. In the absence of reaction terms and, hence, of any evolution process in the Hele-Shaw problem, the limit pressure turns out to be constantly equal to 0, p 1 Á 0. 

Introducing non-trivial Dirichlet boundary conditions drastically changes the behavior of the limiting free boundary problem. In fact, the limit pressure no longer vanishes and this triggers the evolution of the interface in accordance with Darcy's law (which states that the velocity of the free boundary is proportional to the pressure gradient). This problem was addressed in [15], where the authors study the incompressible limit of the porous medium equation defined in OE0; 1/

, where is a compact subset of R d, and the pressure satisfies p. t; x/ D f. t; x/ on @ for some f. t; x/ 0. In the absence of Dirichlet boundary data, i. e. f Á 0, and for large enough, the problem is actually the same as in [5] and it still holds that n 1 D n 1. x/ as well as p 1 Á 0. On the other hand, if one imposes the pressure to be strictly positive somewhere on @, i. e. f 6 Á 0, then the pressure gradient no longer vanishes and the dynamics of the limit problem is governed by Darcy's law. 

The same non-stationary effect, although due to different dynamics, is produced by a non-trivial reaction process. The incompressible limit for (1) without convective effects, i. e. V D 0, and with a pressure-dependent growth rate g D G. p/, was first addressed in the seminal paper [25] by Perthame, Quirós, and Vázquez. They prove that it is possible to extract subsequences of n and p which converge in the L 1 -norm to functions

satisfying the following equation in the sense of distributions on. 0; T / R d, @n 1 @t p 1 D n 1 G. p 1 /; and the relations. 1 n 1 /p 1 D 0 almost everywhere, as well as

in the sense of distributions. The last equality is usually referred to as the complementarity relation and represents the link between the limit equation and the free boundary problem. In fact, denoting by. t / WD ¹x 2 R d j p 1. x; t / > 0º the region occupied by the tumor, from (7) one can see that the pressure satisfies an elliptic equation in the evolving domain. t / with homogeneous Dirichlet boundary conditions. The free boundary @. t / is moving under Darcy's law, which finally allows the fully geometrical representation of the limit problem to be obtained. A derivation of the velocity law can be found in [25] for initial data given by characteristic functions of bounded sets, although the proof relies on formal arguments. A weak (distributional) and a measure-theoretic interpretation of the free boundary condition have been recovered in [23], while in [20] the same result is achieved through the viscosity solutions approach. 

An analogous result regarding the limit ! 1 has been shown in [19] for the pressure law given by (3). The authors obtain virtually the same limiting problem, the only difference being that the complementarity relation (7) becomes [19, Theorem 2. 1]. Let us point out that due to uniform estimates in L 1 the convergence of the sequence of densities is also true in any L p -space, p < 1. 

The Hele-Shaw limit for the porous medium equation including convective effects, cf. (1) with V 6 Á 0, and possibly reaction terms, has attracted a lot of interest as well. Similarly to the driftless case, when passing to the limit ! 1, the model converges to a free boundary problem where, however, the interface dynamics is no longer driven only by Darcy's law, but also by the external drift, i. e. the normal velocity is given by. rp 1 C rV /, where is the outward normal direction. The asymptotics as ! 1 has been addressed both for local and non-local drift, in the absence of reactions; see for instance [1, 9], where the authors adopt techniques relying on the gradient flow structure of the equation. In [21], Kim, Požár, and Woodhouse also include a linear reaction term in the equation and are able to prove convergence to the incompressible limit using viscosity solutions. Recently, in [11] the authors show that the complementarity condition including a drift, i. e. 

holds in the sense of distributions. 

In recent years, many other variations of the model at hand have been proposed, together with the analysis of their incompressible limit. We refer the reader to [10] for a model including the effects of nutrients, [17] for the generalization of the driftless model with a non-monotone proliferation term, and [28] for the model including active motion. In order to account for viscoelastic effects, several models propose to use Brinkman's law instead of Darcy's law [26]. Moreover, cross-reaction-diffusion models using Darcy's law, Brinkman's law, or a singular pressure law have attracted a lot of attention as they raise challenging questions both on the existence of solutions and their incompressible limit; see [4, 6, 13, 14, 18, 22]. 

Our aim is to compute the rate of convergence of the solutions of (1) as " ! 0 or ! 1 in (3) or ( 2 ) respectively. To the best of our knowledge the only result in this direction is given by Alexander, Kim, and Yao [1] for the porous medium equation including a space-dependent drift. Passing to the incompressible limit, the authors are able to build a link between the Hele-Shaw model and the congested crowd motion model

with the constraint n Ä 1. To prove the equivalence of the two models, they study the convergence as ! 1 of the porous medium equation with drift; cf. ( 1 ) with G Á 0. 

Unlike [25], their approach is based on viscosity solutions. On the one hand, they are able to prove locally uniform convergence of the viscosity solution of (1) to a solution of the Hele-Shaw model. On the other hand, they show the convergence of the porous medium equation with drift to the aforementioned crowd motion model in the 2-Wasserstein distance. Therefore, they prove the equivalence of the two models in the special case of initial data given by "patches", namely n 0 D 1 0 for a compact set 0. In fact, the locally uniform limit holds only for solutions of the form of a characteristic function, while the limit in the 2-Wasserstein metric holds for any bounded initial data, 0 Ä n 0 Ä 1 with finite energy and second moment. Moreover, while the local uniform convergence only requires a strict subharmonicity assumption on the drift term, i. e. V 2 C 2. R d /; V > 0, stronger regularity is needed to pass to the 2-Wasserstein limit. More precisely the authors make the following assumptions on V D V. x/: there exists 2 R such that inf

Under these assumptions, they derive the rate of convergence (cf. [1, Theorem 4. 2. ])

where C is a positive constant depending on R V n 0, kV k 1, and T. The main result of this paper offers an improved polynomial rate of convergence in a negative Sobolev norm and the strong topology of Lebesgue spaces; see Theorems 1. 1 and 1. 2 above and Corollary 1. 7 below. Let us remark that the 2-Wasserstein distance and the P H 1 -norm can be bounded by each other when the densities are uniformly bounded away from vacuum: see Appendix A. We refer the reader to [27, Section 5. 5. 2], and references therein, for further discussion of the equivalence of the two distances. 

Preliminaries and assumptions. Throughout this paper we make the following assumptions on the components of the model. Firstly, we assume that (1) is equipped with nonnegative initial data n 0 (resp. n 0 " ) such that there is a compact set K R d and a function

Note in particular that the compact support assumption is needed only in the power law pressure. This is because when the pressure is given by ( 3 ) we can achieve our main estimate without a uniform bound for the pressure in L 1, which is not the case for the power law. Having uniformly compactly supported data allows a maximum principle for the equation satisfied by the pressure to be derived. When additionally specified, we assume further

uniformly in. Secondly, the chemical concentration potential V is assumed to satisfy

and additionally

rV 2 L 1. . 0; T / R d /:

Thirdly, we assume the proliferation rate g D g. t; x/ to be locally integrable and satisfy either g C 2 L 1. . 0; T / R d / and g 0;

where f C WD max. f; 0/ denotes the positive part of the function, or

where f WD max. f; 0/ denotes the negative part of the function, or alternatively

Assuming in addition (A-BVdrift) we also have n 2 L 1. 0; T I BV. R d //. When the pressure is given by (3), points (2) and (3) still hold, and moreover 0 Ä n " Ä 1. 

These bounds are enough for our purposes. Their proofs are fairly standard and derived in full detail in [11, 17, 19, 25], so we omit them here. Let us point out that to fully justify passing to the incompressible limit ! 1, one usually needs to derive additional estimates for the time derivative of the population density and the pressure. 

Remark 1. 5 (More general drift term). It is easily seen in the proof of our main results that we do not require the drift velocity to be a gradient. Indeed, one can replace the term nrV in (1) by nU. t; x/ with appropriate modifications to the regularity assumptions (A-drift) and (A-BVdrift). 

Our approach is to first obtain a rate of convergence in the homogeneous negative Sobolev norm P H 1 and then interpolate with the uniform bound in BV to deduce a convergence rate in Lebesgue spaces. To realize this program we make use of the diffusion structure of the problem and "lift" the Laplacian. More precisely, we define the function ' to be the solution of the following Poisson equation in. 0; T / R d,

given by the convolution ' D K ? n, where K is the fundamental solution of the Laplace equation. Explicitly, for x ¤ 0,

where ! d denotes the volume of the unit ball in R d. Suppose for now that d 3. 

If d D 2, then we do not have ' 2 L 1. R 2 / and we cannot apply Young's inequality to deduce square integrability of r' (indeed, this is an endpoint case). However, let us point out that, for the power law case, since by Lemma 1. 4 (1) solutions are always compactly supported, we can take ' to be the solution of the Poisson equation on K. T / R 2 with homogeneous Dirichlet boundary conditions. In this case, we know that r' 2 L 2. K/. 

Under suitable conditions it is possible to infer the L 2 -integrability of r. ' " ' " 0 / in R 2, which is needed for the singular pressure law. In this case, we impose the following additional assumptions:

The bound on the first moment is propagated in time and guarantees the well-posedness of K ? n ". Taking a space-independent growth rate implies that the difference n " n " 0 has zero mean for all times. Therefore, we have

from which we conclude that r. ' " ' " 0 / 2 L 2. R 2 /: Notice that the L 1 convergence of the initial data implies the convergence of r' 0 to r' 0 1 in L 2. Moreover, the uniform bounds on n together with the Hardy-Littlewood-Sobolev inequality imply that the convolution n 7 ! K ? n is a bounded linear operator from L 2d=d C2 to L 2. Therefore there is a subsequence r' k which converges weakly in L 2 to r' 1. 

Finally, we recall that the gradient r' can be used to represent the P H 1 -norm of the function n as

Having obtained a convergence rate in the negative norm and assuming additionally the BV bounds provided by Lemma 1. 4, we will use the following interpolation inequality, proved (in greater generality) by Cohen et al. [8] (see also [7] ) to deduce a rate in the Lebesgue 4=3-norm: Lemma 1. 6 (Interpolation inequality). There exists a constant C D C. d; T / > 0 such that, for all t 2 OE0; T,

Thus, Theorem 1. 2 is a simple consequence of Theorem 1. 

In particular, this covers the case of "patches", i. e. when the initial distribution is given by an indicator function of a compact set, as considered recently in [1]. 

Plan of the paper. The remainder of the paper is devoted to proving the main theorem. It turns out that the equation can be conveniently trisected and dealt with term by term: the pressure-driven advection, drift, and proliferation are considered separately. Indeed, it is the diffusion term that governs the rate of convergence. The proof is therefore structured as follows. In Sections 2 and 3 we prove the main theorem for the choice of the singular pressure in (3) and the power law pressure in (2) in the absence of reactions and drift. 

Then in Section 4 we explain how to treat the additional terms. 

Notation. Henceforth we shall usually suppress the dependence on time and space of the quantities of interest, only exhibiting the time variable in the final results. Similarly, for the sake of brevity, all space integration should be understood with respect to the ddimensional Lebesgue measure.
section_name: Introduction

section_content: In this and the following section, to explain the main idea in a simple situation, we ignore the drift and proliferation terms in (1) and consider only the nonlinear diffusion equation @n " @t r. n " rp " / D 0; (10) assuming now the pressure law as in (3). In this case we can rewrite (10) as @n " @t H ". n " / D 0; with

Recall that we have the uniform bound n " < 1, so that the right-hand side above is well defined with ln. 1 n " / Ä 0. Let us take " > " 0 > 0. We subtract the equation for n " 0 from the equation for n " to obtain @. n " n " 0 / @t. H ". n " / H " 0. n " 0 // D 0:

Now we pose (8) for both solutions n " and n " 0 :

Then (11) reads @. ' " ' " 0 / @t. H ". n " / H " 0. n " 0 // D 0;

and we test it against ' " ' " 0 to derive

. n " n " 0 /. H " 0. n " 0 / H ". n " //:

We now proceed to estimate the right-hand side. On the set ¹n " > n " 0 º we make use of the non-negativity of H ". n " / and non-positivity of the logarithmic term in H " 0. n " 0 / to write

Similarly, on the complementary set ¹n " Ä n " 0 º we have

n " :

Therefore we have

and since n " and n " 0 are uniformly bounded in L 1. . 0; T /; L 1. R d // with respect to " and " 0, we obtain 1 2

Integrating in time on OE0; t / we then have

It follows that the sequence. r' " / " converges in the strong topology of L 1. . 0; T /; L 2. R d // to r' 1. Consequently, letting " 0 ! 0, we deduce the following rate for the convergence n " ! n 1 in the space P H 1. R d /:

where C is a positive constant defined as

Assuming the additional BV bounds for the initial data, we get from Lemma 1. 4 that n " is uniformly bounded in L 1. 0; T I BV. R d //, and we can use (9) to obtain the rate " 1=4, as announced in (4). Thus Theorems 1. 1 and 1. 2 are proved in this special case.
section_name: Singular pressure law

section_content: Let us now consider (10) with the pressure law given by ( 2 ) and demonstrate that the method employed in the previous section remains valid. We now have the porous medium equation @n @t n D 0:

Let us recall that there exists a positive constant p M such that

Then it immediately follows that Q n Ä 1 and solves the equation

Following the same argument as before, we define ' and Q ' by

Without loss of generality, we take 1 < < 0. Now we subtract the equation for Q n 0 from the equation for Q n to obtain

Then from (12) we have

where the inequality follows from the fact that Q

It is easy to see that for 0 Ä s Ä 1 it holds that s. 1 s/ Ä s. Hence, we have

where in the last inequality we used the fact that by Lemma 1. 4, n is uniformly bounded in L 1. 0; T I L 1. R d //. Finally, we remove the scaling using the triangle inequality:

By the definition of c, j1 1 c j 2 ! 0 as ! 1. Thus, we have kr. '

By the same argument, we find

Finally, we conclude kr. '

Consequently, arguing as before and letting 0 ! 1, we find

Again, under the additional BV assumptions we obtain (4) thanks to the interpolation inequality in Lemma 1. 6.
section_name: Power law

section_content: Having obtained the announced rate of convergence due to the nonlinear diffusion term, we now show that we can include the drift and reaction terms. In fact, due to our assumptions on the proliferation rate and the chemical potential, all the additional terms will either have an appropriate sign, or be absorbed into the L 2 -norm of the potential '. We now write (1) as @n @t A. n / D r. n rV / C n g;

where g D g. t; x/ and A is chosen appropriately depending on the state law for the pressure. As seen before, there is no harm in assuming the uniform bound n Ä 1. Then, arguing in the same way as previously, we obtain

It only remains to consider the two new terms on the right-hand side. For the first one we can write

where we have integrated by parts and used assumptions (A-drift). For the remaining term we integrate by parts to obtain

In case of d D 2, we suppose that g satisfies assumption (A-reaction). Then we can integrate by parts in the last term to obtain

If instead d 3, we may alternatively assume that g satisfies assumption (A-reaction 0 ) or assumption (A-reaction 00 ). In the first case, successively using the Hölder and Sobolev inequalities, we obtain

where C S denotes the constant from the Sobolev inequality, and 2 D 2d d 2 is the Sobolev conjugate exponent. Otherwise, if g satisfies (A-reaction 00 ), to estimate the term A we do not integrate it by parts, but we use in turn the Young, Hölder, and Sobolev inequalities, to obtain

Therefore we have

Assuming for concreteness the power law pressure, using inequality ( 14 ) and a Grönwall inequality, we deduce sup t2OE0;T kr. '

Finally, passing to the limit 0 ! 1, we conclude the proof of Theorem 1. 1. Using the uniform BV-bound and ( 9 ) we obtain Theorem 1. 2.
section_name: Including drift and reaction terms

section_content: Here we prove relation ( 5 ) between the limit density and pressure, where p 1 is defined as the weak limit (up to a subsequence) of p in L 1. . 0; T / R d /. 

Proof of Theorem 1. 3. The relation is a straightforward consequence of the main estimate obtained in Section 3. We inspect (13), this time not ignoring the non-positive terms. After integration in time, using ( 14 ) these terms can be bounded as

Now let be a compactly supported test function and consider the quantity

Using weak lower semicontinuity of convex functionals and weak convergence of the pressure and the density, we can pass to the limit with 0 and in turn to obtain

which concludes the proof.
section_name: Limit relation between n 1 and p 1

section_content: We computed the rate of convergence of the solutions of a reaction-advection-diffusion equation of porous medium type in the incompressible limit. Our result in a negative Sobolev norm can be interpolated with uniform BV-estimates in order to find a rate in any L p -space for 1 < p < 1. 

How to assess the accuracy of our estimate remains an open problem. For the pure porous medium equation it might seem tempting to attempt a calculation for the illustrious example of the Barenblatt solution (taking as initial data the solution at some time t > 0). However, a direct calculation shows that in this case the data is "ill prepared" in the sense that it converges (in L 1 ) to its limit profile with too slow a rate of ln =. It is unclear how to approach the question of optimality in general. We expect that the "worst" rate would be exhibited by a focusing solution, whose support is initially contained outside a compact set and closes up in finite time, thus generating a singularity. 

Another challenging problem is to find an estimate for the convergence rate of the pressure, for which the method used above seems inapplicable as it is not clear how to relate the quantities p p 0 and ' ' 0. Consequently, we are also currently unable to treat more general, pressure-dependent, reaction terms. Finally, it would be of interest to investigate whether it is possible to strengthen the estimate of Theorem 1. 1 to Lebesgue norms without interpolation with BV. One advantage of any such alternative approach could be to allow for passing to the incompressible limit when BV bounds are not available, as is the case for systems of equations like (1). Additionally, it could allow for estimating the rate of convergence in the L 1 -norm rather than the seemingly arbitrary L 4=3 -norm. 

A. Bounding the W 2 -norm by the P H 1 -norm

We consider here the conservative case of (1), assuming R n. t / D R n 1. t / D 1. Moreover, rather than the Cauchy problem set in the whole space, we consider the boundary value problem set in a bounded domain R d with homogeneous Neumann boundary conditions. 

We put d D n. x/ dx, d 1 D n 1. x/ dx, ignoring time dependence for the sake of brevity. Furthermore, we make the additional assumption that n 1 n > 0 for some constant n. 

Consider the curve W OE0;
section_name: Conclusions and open problems

section_content: the European Union's Horizon 2020 research and innovation program(grant agreement no. 740623). T. D. was partially supported by the National Science Center (Poland), grant number 2018/31/N/ST1/02394, and the Foundation for Polish Science (FNP).