From the highly compressible Navier-Stokes equations to the Porous Medium equation - rate of convergence

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\frac{1}{\sqrt{\varepsilon}}$ for $\varepsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation-$\rho_\varepsilon$ converges to the unique solution to the porous medium equation [13,14]. For viscosity coefficient $\mu(\rho_\varepsilon)=\rho_\varepsilon^\alpha$ with $\alpha>1$, we obtain a rate of convergence of $\rho_\varepsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<\alpha\leq\frac{3}{2}$ the solution $\rho_\varepsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $\rho_\varepsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\varepsilon$.

1. Introduction. The compressible Navier-Stokes equations in the multidimensional case with the Mach number equal to ε −1/2 read: where ε = ε (t, x) and u ε = u ε (t, x) denote the unknown density and the velocity vector field, respectively, P ( ε ) = γ ε , γ > 1, denotes the pressure, D(u ε ) = 3108 BORIS HASPOT AND EWELINA ZATORSKA Our purpose in this paper is to study the asymptotic behaviour of the global weak solutions to the one-dimensional Cauchy problem for (1), when ε goes to 0, which corresponds to the highly compressible limit. It is the opposite to the low Mach number limit leading to the incompressible Navier-Stokes equations. The latter has been intensively studied, especially in the case of constant viscosity coefficients. The first works in the framework of global weak solutions are due to P.-L. Lions and N. Masmoudi [25] and B. Desjardins and E. Grenier [9] (see also [10]). The case of global strong solution with small initial data was considered by S. Klainerman and A. Majda in [21] for well-prepared initial data and more recently in critical spaces for scaling of the equations by R. Danchin for ill-prepared initial data in [8]. Up to our knowledge the highly compressible regime has not been so well studied. The only results we are aware of concern Euler equations, see for example [7], [12]. The main difficulty to pass to the limit ε → 0 in (1) with constant viscosity coefficients is due to lack of uniform estimates on { ε } ε>0 (only the L 1 norm is conserved). In this case, it is not clear whether the sequence of weak solutions to (1) converges to solution of the pressureless system. However, in the case of viscosities satisfying particular algebraic relation additional compactness information is available. The first author proved in [14,13] that there exists a limit of the sequence (ρ ε , u ε ), called a quasi-solution, satisfying the pressureless system when N ≥ 2. Moreover, the density of this system corresponds to the solution of the fast diffusion, the heat or the porous medium equation, depending on the choice of the viscosity coefficients. Relation (2) introduced by D.
Bresch and B. Desjardins in [1,2] provides a new entropy structure which ensures estimate on the gradient of the density. In particular, it gives enough compactness information to pass to the limit when ε goes to 0. Stability of global weak solutions to (1) with (2) for ε fixed was proved by A. Mellet and A. Vasseur [26] using new energy estimate improving the integrability of the velocity. A particular choice of viscosity coefficients µ( ) = , λ( ) = 0 satisfying (2) leads to the so-called shallow water system, for which the proof of existence of global weak solutions has been recently delivered by A. Vasseur and C. Yu [30], the extension to the case (2) can be found in [31], see also [11,17]. Some ideas on construction of solution using the cold pressure (a pressure that is singular at the vacuum) can be found in [3,29,38] and in [4] using the notion of the κ-entropy solutions.
In the following paper, we extend [14,13] by proving rate of convergence of ε to a solution of the porous medium equation in a suitable functional framework. We restrict to the one-dimensional Euclidean space R, for which system (1) may be rewritten with a single general viscosity coefficient µ( ) > 0 in the form Following [15] (see also [16] and [4]), we rewrite the above system in terms of the effective velocity THE COMPRESSIBLE N-S EQUATIONS TO THE PM EQUATION   3109 which leads to a appearance of a parabolic term in the continuity equation This new regularising effect was observed in [16] in the framework of strong solution corresponding to initial density far away from the vacuum. The same concept was recently used in [5,4] to construct regular solutions approximating global weak solutions to the full low Mach number limit system and to the compressible system. Our starting observation is that when v ε (0, ·) = 0, the sequence {ρ ε v ε } ε>0 converges to 0 in a suitable functional framework. It heuristically implies that the limit ρ = lim ε→0 ρ ε solves the fast diffusion, the heat or the porous medium equation where µ(ρ) =ρ α for α > 0.
The rate of convergence of (ρ ε −ρ) can be obtained by employing a duality method from [33], used by J. L. Vázquez to prove uniqueness of the very weak solutions to the porous medium equation.
In the present paper we restrict to the case α > 1 because we are particularly interested in dealing with compactly supported initial data. When 0 < α < 1 2 the entropy discovered by D. Bresch and B. Desjardins (see [2]) allows to bound the density from below. Indeed, in [27] A. Mellet and A. Vasseur proved the existence of global strong solution to (3) for initial density far away from the vacuum. The main ingredient of the proof was to use entropy from [2] to estimate ∂ x (ρ α− 1 2 ) in L ∞ (0, T ; L 2 (R)) for all T > 0. This implies boundedness of ρ −1 in L ∞ (0, T ; L ∞ (R)) which allows to show sufficient regularity of solution in order to prove the uniqueness. This result has been recently extended by the first author to the case of the shallow water system (α = 1) [15]. The main idea was to prove boundedness of v ε appearing in (4) by using the structure of the transport equation for v ε . It allows to bound ρ −1 by application of the maximum principle to the continuity equation.
Finally, let us mention several important results concerning existence of global weak and strong solutions to system (3) with initial density close to vacuum. Presenting the exhaustive overview of the theory in this field is however beyond of the scope of this article.
The existence of global weak solutions to system (3) was proven by Q. Jiu and Z. Xin in [20]. The main difficulty was to construct approximate regular solutions to (3) which verify all the entropies used in [26]. To construct such an approximation, Q. Jiu and Z. Xin employed the result on existence of global strong solution on bounded domain [−M, M ] with Dirichlet boundary conditions. The latter result was proven by H-L. Li., J. Li and Z. Xin [24] who used energy estimate to control the Lipschitz norm of the velocity.
There are also several interesting papers on system (3) with free boundary corresponding to the interface with the vacuum. The first result in this direction is due to D. Hoff and D. Serre [18] who proved the existence of global strong solution with discontinuous density at the interface for constant viscosity coefficients. An interesting extension of this result to the Neumann boundary conditions is due to P. B. Mucha [28] and for the reactive system see [23]. S. Jiang, Z. Xin and P. Zhang in [19] obtained similar results in the case of degenerate viscosity coefficients. The initial density considered in this paper is discontinuous at the interface and therefore is not compatible with the entropy introduced in [2]. We shall discuss this problem with more details later, in Section 5.
The free boundary problem with initial density continuously connecting to the vacuum was analysed by T. Yang and H. Zhao in [36]. They proved the existence of weak solution in a finite time interval in Lagrangian coordinates by showing that for sufficiently small T the L ∞ (0, T ; L ∞ (Ω)) norm of ρ α+1 ∂ x u is bounded. Unfortunately, this is not sufficient to come back to the Eulerian description since it is not clear how to solve the ordinary differential equation describing the evolution of the free boundary. In other words, the change of coordinates may not be well defined, see also [35,37,34].
The article is structured in the following way. In Section 2 we formulate the notion of the weak solution to problem (3) for ε fixed and we state our main results. Next, in Section 3 we recall the result from [20] on existence of global in time weak solutions and derive the estimates independent of ε. In the second part of this section we recall basic properties of the solution to the one-dimensional porous medium equation taken over from the monograph of J.-L. Vazquez [33]. In Section 4 we prove the main results of the paper and in the end, in Section 5 we explain difficulties arising in the study of the free boundary problems and discuss possible extensions of our results.
Notation. The letter C denotes generic constant, whose value may change from line to line.

2.
Main results. Below we give the definition of a global weak solution to problem (3) which we denote by (ρ ε , u ε ) or by (ρ, u) when no confusion can arise. The existence of such solution was proven by Q. Jiu and Z. Xin in [20]. We then formulate our main theorems in which we compare ρ ε withρ the solution to the porous medium equation.
Let us start with introducing the hypothesis on the initial data, we assume that where α > 1 2 and ν > 0 arbitrary small. Definition 2.1. Let ε be fixed. We say that a pair of functions (ρ, u) is a weak solution to system (3) provided that: • The density ρ ≥ 0 a.e., and the following regularity properties hold where (W 1,∞ (R)) is the dual space of W 1,∞ (R). • For any t 2 ≥ t 1 ≥ 0 and any ψ ∈ C 1 ([t 1 , t 2 ] × R), the continuity equation is satisfied in the following sense: Below we give our first main result on the convergence of (ρ ε , u ε ) to a solution to the associated porous medium equation.
In addition, ρ ε converges strongly toρ -the strong solution to the porous medium equation in the following sense, there exists a constant C > 0 depending on ρ 0 such that 2. For 1 < α ≤ 3 2 there exists a constant C > 0 depending on ρ 0 such that for all t ≥ 0.
Remark 1. As mentioned above, the existence of weak solutions to system (3) was proven by Q. Jiu and Z. Xin in [20]. The novelty here is that our definition of weak solution incorporates additional equation (9).
Remark 2. The initial data from (12) can be replaced by a condition which imposes this equality only approximately: with η > 0. It implies that the inequality (14) is true up to an error term coming from (16) for a constant C > 0 depending only on the initial data ρ 0 . The same applies to inequality (15).
The next main result is the following.
Assume that the initial conditions satisfy (7) and that there exist two constants −∞ < a < b < +∞ such that Then, there exist constants C > 0 and −∞ < a 1 < b 1 < ∞ such that Moreover, there exists a constant C > 0 depending only on ρ 0 such that where Ω c t denotes the complement of Ω t . Remark 4. The finite speed of propagation of the interface (18) is a classical result following from the theory of Porous Medium equation.
Remark 5. Theorem 2.2 provides a rate of convergence of (ρ ε −ρ) and so it extends the results of the first author proven in [14,13] when N ≥ 2. Theorem 2.3 implies that the mass distributed outside Ω t -the support of solution to the porous medium equation, is small of order ε 1 4 . The evolution of the support of Ω t is known since the interfaces are described by the Darcy law (see Theorem 3.5). However, ρ ε may not be a solution to the free boundary problem. In other words, there is no reason for suppρ ε (t, ·) to remain compact for all times t > 0. However, the mass that may be spread outside of Ω t remains small. In this senseρ(t, ·) is a good approximation of ρ ε (t, ·) (for any t ≤ max(1, cε −1/(1+ 2 1+α ) ) for c small) and we can think about the "quasi finite propagation" of the mass.
3. Overview of known results. This section is devoted to the overview and summary of results that will be used to prove our main theorems.
3.1. Weak solutions to system (3). We first recall the result on global in time existence of weak solutions to system (3) with ε being fixed, whose proof can be found in the paper of Q. Jiu, Z. Xin ( [20], Theorem 2.1.) Theorem 3.1 (Existence of weak solutions). Let γ > 1, α > 1/2 and let ε > 0 be fixed. Assume that the initial conditions satisfy (7). Then, system (3) possesses a global in time weak solution (ρ, u) in the sense of Definition 2.1. Moreover, this solution satisfies the following inequalities uniformly with respect to ε sup sup where c depends only on the initial data and Λ ∈ L 2 ((0, T ) × R) is a function satisfying: Proof. The only element that we have to check is verification of the artificial formulation of the continuity equation (9). The rest was already proven by Q. Jiu, Z. Xin in [20] (see Theorem 2.1). Their proof combines the existence of global solutions on bounded domain [−M, M ], proven in [24] with the diagonal procedure allowing to let M → ∞. Therefore, we omit the details of construction of this approximate solution and focus only on justification that the formula (9) is valid. Assume that ρ δ,0 , m δ,0 converge to ρ 0 , m 0 in the following sense ρ δ,0 → ρ 0 , strongly in L 1 (Ω), We show that (9) is weakly sequentially stable when δ → 0. To this purpose let us recall the a-priori estimates derived in [24] following the strategy developed by A. Mellet and A. Vasseur in [26] for the multidimensional case. First recall that the classical energy balance gives for every T > 0 the following inequality

BORIS HASPOT AND EWELINA ZATORSKA
In addition, the BD entropy gives rise to the estimates where we denoted ϕ δ (ρ δ ) = µ δ (ρ δ ) . Therefore, we obtain the following bounds where the constant is independent of δ.
Following [24] we obtain This estimate can be then used to derive the improved estimate for the velocity, exactly as in [26]. Testing the momentum equation by u δ |u δ | ν for ν > 0 sufficiently small, and testing the continuity equation by |u δ | 2+ν 2+ν and summing the obtained expressions we obtain Due to the boundary conditions the r.h.s. can be integrated by parts and estimated as follow Note that due to our assumptions 2γ − α > 0. For r sufficiently small, the first term can be absorbed by the l.h.s. of (29), while to control the second term, we can use the natural bounds for (27) and (28) Having these estimates, we can follow the compactness arguments from [26] and from [24] to prove the convergence of the approximate solution (ρ δ , u δ ) to some weak solution (ρ ε , u ε ) as δ → 0. Namely where Λ satisfies (23). At this point, sequential stability of equations (8) and (10) is verified. To justify the limit passage in (9), one has to prove the convergence of the term ρ δ v δ . Observe that we have Due to (30) the first term converges to ρ ε u ε strongly in L 2 (0, T ; L 2 (Ω)), while the second term converges to 2 2α−1 ρ weakly in L 2 (0, T ; L 2 (Ω)); the last one converges to 0. In consequence and so one can pass to the limit in (9). This is the final argument to prove the sequential stability of weak solutions to (4) on the bounded domain Ω = [−M, M ] and with the Dirichlet boundary condition for u δ . In order to let M → ∞, we combine the diagonal procedure with the convergence of the initial data (24) as it was done in [20].

The porous medium equation.
In order to understand the qualitative properties of the limit solution to (3), we recall several important features of the porous medium equations. The majority of them is taken from the excellent books of J. L. Vázquez [33], [32]. The porous medium equation can be written as follows: with α > 1.
In the sequel we shall set Q = (0, +∞) × R. Let us recall the notion of global strong solution for the porous medium equation (31) (see [33] Chapter 9 for more details).
Definition 3.2. We say that a functionρ ∈ C([0, +∞), L 1 (R)) positive is a strong L 1 solution to problem (31) if: The existence of global strong solution to (31) is guaranteed by the following theorem (see [33] page 197).
Let ρ 1 and ρ 2 be two strong solutions of (31) in (0, T ) × R then for every 0 ≤ τ < t If ρ 1 and ρ 2 are two strong solution with initial data (ρ 1 ) 0 and (ρ 2 ) 0 , such that Likewise above, there exists a theory of global weak solutions with initial data being bounded measures. It covers the case of very important case of self similar solutions, the so called Barenblatt solutions U (t, x, M ), of the form where κ = α−1 2α(α+1) and C > 0 with C = cM 1 γ with γ = α+1 2(α−1) and c depends only on α > 1.
The Barenblatt solutions verify (31) for t > 0 in the sense of distributions and with initial dataρ 0 = M δ 0 , where δ 0 is the Dirac mass.
Analogue solutions exist also in the case of fast diffusion equations corresponding to (31) with α < 1.
Remark 6. The comparison principle from Theorem 3.3 ensures a finite speed of propagation of solutions to the porous medium equation with compactly supported initial data, see for example [6,33]. In other words, the solution to the porous medium equation with compactly supported initial data remains compactly supported all along the time. Indeed, it suffices to compare such solution with the Barenblatt solutions.
Let us now recall the so called L 1 − L ∞ smoothing effect (see [32] page 202).
Theorem 3.4. Letρ 0 ∈ L 1 (R). For every t > 0 we have: We recall an important theorem (see [33] page 376) which gives a precise behaviour of the interfaces whenρ 0 has compact support.
Theorem 3.5. Let x = s(t) denote the right interface of the solutionρ to (31) emanating from the compactly supported initial dataρ 0 ∈ L 1 (R). For all t > 0 there exist the one side limits Moreover, the Darcy law holds in the form s(t)). The same result is true for the left interface.

4.
Proof of main results. Below we present the proofs of our main results stated in Theorem 2.2 and Theorem 2.3. We start by proving the rate of convergence from (15), the convergence of the approximate solutions (ρ ε , u ε ) to the solutions of the corresponding porous medium equation is a consequence of this estimate, the details can be found in [14].
Proof of Theorem 2.2. First let us note that thanks to Theorem 3.1, the continuity and it is satisfied in the sense of distributions. From (12) and (26) it also follows that Next, we considerρ -the solution to the corresponding porous medium equation with the same initial data ρ 0 whose main properties were recalled in Section 3.2. We now set It follows from (35) and (37) that R ε satisfies the equation at least in the sense of distributions.
Our goal is to estimate a relevant norm of R ε in terms of ε. To this purpose, we use duality technic in the spirit of J. L. Vázquez (see [33] Section 6.2.1). Testing (38) Let us now define This definition implies in particular that ρ α ε −ρ α = aR ε . We can hence rewrite (39) as The next step consists of solving the dual problem in the interval where θ ∈ C ∞ 0 ((−M, M )). Above, a n is a smooth approximation of a such that 0 < η ≤ a n ≤ K < +∞ (it will be precisely defined later on). This assumption guarantees that system (41) is parabolic in the reverse time t = T − t and it admits unique smooth solution ψ on [0, T ].
Since ψ is a solution to (41), we can use (40) to obtain We now need to obtain the a priori estimate for ∂ xx ψ. To this purpose, we multiply equation (41) by ζ∂ xx ψ, where ζ = ζ(t) is a smooth and positive function such that Integrating the first term by parts and using the fact that ψ(t, ·) ∈ C ∞ 0 ([−M, M ]) with all derivatives belonging to C 0 ([−M, M ]) (see the general theory of [22], page 341), we obtain Hence, it follows that Plugging it into (43) we obtain

FROM THE COMPRESSIBLE N-S EQUATIONS TO THE PM EQUATION 3119
In particular, the assumptions on ζ imply that we have Coming back to (42) [a − a n | 2 a n |R ε | 2 dx dt (47) At this stage following ideas from [33] we construct the approximation a n that is sufficiently regular and verifies η ≤ a n ≤ K for η > 0 small and for K > η large enough. This requires two steps of approximation. The first consists of taking a K,η = min(K, max(η, a)) with 0 < η < K, in other words Since a K,η is L ∞ ((0, T ) × R), we know that a K,η is also in L p ((0, T ) × (−M, M )) for 1 ≤ p ≤ +∞. Therefore, in the next step, we can consider a smooth approximation a n = a K,η * ψ n (with ψ n a standard regularising kernel) such that for n → ∞ a n → a K,η strongly in L p ((0, T ) × (−M, M )) for all 1 ≤ p < +∞. It remains to estimate the r.h.s. of (47). Since a K,η ≥ η we deduce that also a n ≥ η for any n, therefore (48) We call the two integrals on the r.h.s. I 1 and I 2 , respectively. The integrand of I 2 is pointwise bounded by (21) and (7)), using the dominated convergence theorem we justify that where we used the fact that R ε is uniformly bounded in L ∞ ((0, T ) × R) for 0 < T < +∞.
To estimate I 1 , it suffices to observe that R 2 ε is actually uniformly bounded in ε in L p ((0, T ) × R) for any 1 ≤ p ≤ +∞ due to (21) and due to L ∞ bound on the solution to the porous medium equation. We now use the fact that a n → a K,η when n → +∞ strongly in L p ((0, T ) × (−M, M )) to obtain that T 0 M −M |a − a n | 2 |R ε | 2 dx dt → n→+∞ 0. Using this, (49) and (48) we deduce that for n = n(η, K, M, T ) large enough we have: Coming back to (47) and using the Hölder inequality we obtain that there exists C > 0 such that for every η > 0: From (28) and (36) we thus deduce that To conclude, let us observe that the above inequality holds for any θ from C ∞ 0 ((−M, M )) with arbitrary M > 0 and that the constant C is independent of M . Thus, letting M to +∞ we obtain for any T > 0.
Proof of Theorem 2.3. Having proven the above rate of convergence, our next aim will be to provide some more information about the behaviour of ρ ε on the support of the solution to the corresponding porous medium equation.
Under assumptions of Theorem 2.3 and from the properties of the solutions to the one-dimensional porous medium equation presented in Section 3.2, we deduce that the suppρ(t, ·) remains compact for all time t > 0 and it satisfies (18). Indeed, since suppρ 0 is included in [a, b], we can consider the Barenblatt solution U (t 1 , M ) at time t 1 > 0 such that Last inequality is verified if M is large enough depending on ρ 0 L ∞ and 1 t1 (see formula (33)). Using the maximum principle we deduce that:
This finishes the proof of Theorem 2.3.

5.
Where are the interfaces? A natural question arising in the analysis of onedimensional problems with compactly supported initial data and free boundary conditions is propagation of the interface. From the classical theory for porous medium equation (see Section 3.2) we know that the interface moves with a finite speed. Moreover, according to (19) we can estimate the mass corresponding to ρ ε inside the support ofρ. This could suggest that the interfaces of free-boundary Navier-Stokes equations behave similarly to the interfaces of the porous medium equation. However, the issue of existence of global weak solutions to the freeboundary Navier-Stokes equations remains open. Below, we present an overview of partial results in this topic and explain the main problems.
1. A case of compactly supported initial density with jump discontinuity at the interface was studied in [19]. In this article S. Jiang, Z. Xin, and P. Zhang proved existence of global strong solution to (3) written in Lagrangian coordinates. Extending their solution outside the free-boundary domain by imposing the Rankine-Hugoniot condition on the velocity, allows to control ∂ x u ε in L 1 (0, T ; L ∞ (R)). It enables in particular to obtain global strong solution on the whole R. It is also sufficient to define the free boundary which verifies an ordinary differential equation. In this case one can pass from the Eulerian to the Lagrangian coordinates and conversely.
It would be therefore interesting to compare the density ρ ε from [19] with solution to the porous medium equationρ emanating from the same initial data ρ 0 . However, since ρ 0 is discontinuous, it is not compatible with assumptions √ ρ 0 ∂ x ϕ(ρ 0 ) ∈ L 2 (R) and ρ 0 ∈ L 1 (R). As a consequence, we loose the a priori estimate on √ ρ ε v ε (26) which was necessary to show damping of ∂ x (ρ ε v ε ) in (4). In fact, the behaviour of ρ ε is completely different than behaviour of˜ . The discontinuity at the interface of ρ ε exists all along the time, whileρ becomes continuous in arbitrary finite time (see Remark 6). 2. The free-boundary problem with compact initial density, continuously connected to the vacuum was studied in [36]. There, T. Yang, Z-a. Yao, and C. Zhu proved the local in time existence of weak solutions to (3) in Lagrangian coordinates. Although this setting seems to be more adequate for our considerations, the existence of interface corresponding to this solution is not clear. Indeed, in order to determine the evolution of the interface one needs to come back to the Eulerian coordinates. Unfortunately, result obtained in [36] does not guarantee boundedness of the gradient of the velocity in L 1 (0, T ; L ∞ (R)). This lack of control on ∂ x u ε appears at the boundary, where the density vanishes.
For the porous medium equation, the behaviour of the interface in the 1D case is well understood since it verifies the Darcy law (see Theorem 3.5). It could be also appropriate to replace the usual free boundary condition for compressible Navier-Stokes equations by the Darcy law at the interfaces. Indeed in this case we could give a sense to u ε (t, s 1 (t)) (where s 1 denotes the right interface) by considering the limit lim x→s(t) − u ε (t, x) on the left hand side of the free boundary s 1 . However with such condition the Lagrangian change of coordinates is a priori not possible.