A POLYGONAL SCHEME AND THE LOWER BOUND ON DENSITY FOR THE ISENTROPIC GAS DYNAMICS

. Positive density lower bound is one of the major obstacles toward large data theory for one dimensional isentropic compressible Euler equations, also known as p-system in Lagrangian coordinates. The explicit example ﬁrst studied by Riemann shows that the lower bound of density can decay to zero as time goes to inﬁnity of the order O ( 11+ t ), even when initial density is uniformly positive. In this paper, we establish a proof of the lower bound on density in its optimal order O ( 11+ t ) using a method of polygonal scheme.

1. Introduction. In this paper we consider the Cauchy problem for isentropic gas dynamics in Lagrangian coordinates where the specific volume v = 1/ρ, the density ρ > 0 and the velocity u of the gas are all functions of (x, t) ∈ R × R + . The pressure p(v) satisfies where γ is the adiabatic constant, and K is a positive constant. This system is also called the p-system. For Lipschitz continuous solutions, (1) is equivalent to the corresponding Cauchy problem of isentropic Euler equations in Eulerian coordinates [23]. Letting c = −p (v) be the sound speed, the Riemann invariants s and r are defined as For smooth solutions, s and r satisfy s t + cs x = 0 and r t − cr x = 0 .
Toward a large data theory, such as the existences of BV solutions for isentropic Euler equations (1), one of the main challenges comes from the possible degeneracy when density approaches zero. In fact, when solution approaches vacuum, system (1) loses its uniformly strict hyperbolicity, which causes major difficulties in analyzing the large data solutions, see [2,5,18] for analysis and examples showing these difficulties. Therefore, sharp information on the time decay of density lower bound is critical in the study of compressible Euler equations.
It is well known for (1) that the density can be arbitrarily close to zero as time goes to infinity, even when initial density is uniformly away from zero, such as in the interaction of two strong rarefaction waves, c.f. [9,20]. In his pioneer paper [20] of 1860, Riemann studied the interaction between two rarefaction waves. Based on Riemann's construction, when γ = 2N +1 2N −1 with any positive integer N , Lipschitz continuous examples were provided in Section 82 of [9], in which density functions are proved to decay to zero in an order of O(1 + t) −1 . For reader's convenience, a relative detailed discussion can be found in Section 2 below. Similar example can be found in [14] for any γ > 1. In this paper, for 1 < γ < 3, we will prove the sharp positive density lower bound of the order O(1 + t) −1 for any Lipschitz continuous solutions of (1).
Thanks to the elegant structure of Euler equations, the local behavior of smooth solutions can be classified into two classes: compression and rarefaction, as defined below, see for instance [4,8,15,21]. These concepts can be generalized to Lipschitz continuous solutions without any difficulty. Definition 1.1. At any point on (x, t)-plane, the smooth solution is forward (resp. backward) rarefaction R ( R) if and only if s x ≥ 0 (resp. r x ≥ 0) at that point; forward (resp. backward) compressive C ( C) if and only if s x < 0 (resp. r x < 0) at that point. Among many results, two of them are closely related to this paper. For purely rarefactive piecewise Lipschitz continuous solutions, Lin [17] proved that the density has a O(1 + t) −1 order lower bound. For general smooth solutions, including compressions in the solution, in a very recent paper [7], we find a O(1 + t) − 4 (3−γ) lower bound when 1 < γ < 3. This result helps us to prove that gradient blowup of u and/or v happens in finite time if and only if the initial data are forward or backward compressive somewhere. This latter result is further extended in [7] to nonisentropic flow.
In this paper, for general Lipschitz continuous solutions of (1) when 1 < γ < 3, we improve the lower bound on density from O((1 + t) − 4 (3−γ) ) in [7] to the optimal order of O(1 + t) −1 . Based on this result, the estimate in [7] on the life-span for classical solutions of (1) can be improved.
Our main result of this paper is the following Theorem.
Theorem 1.2. Let 1 < γ < 3. Assume that initial data s 0 (x) = s(x, 0) and r 0 (x) = r(x, 0) are uniformly bounded and Lipschitz continuous functions on x ∈ R, and denote that max x =y Furthermore, assume that v(x, 0) in the initial data has uniformly positive upper bound: max For any given positive time T , if u(x, t), v(x, t) is a Lipschitz continuous weak solution on R × [0, T ] for the initial value problem of (1), then there exists a positive Here, L 1 and L 2 measure how rarefactive the initial data are.
To prove this theorem, we study a polygonal scheme similar to those used in [17,19]. The polygonal scheme was first introduced by Dafermos [11] in the study of scalar conservation law, then was modified by Diperna for system of conservation laws [12]. This scheme, under a renowned name front tracking, has been widely used for the well-posedness and behaviors of solutions for hyperbolic conservation laws. For more details on the development of front tracking method, see for instance [1,10,13,22], and references therein.
The main difference of our scheme from the previous ones, is that we add a memory of the local rarefaction and compression character on our scheme, which helps us finally track the maximum rarefaction on waves. And this quantity is crucial in estimating the decay of density. Here, we choose the polygonal scheme since it fits the p-system very well. We wish to point out here that one of our motivations in this paper besides showing the density lower bound is to provide the readers a new method in tracking the variation of solutions during the wave interactions.
In order to track the density changes along the time evolution, it is important to characterize the key players of density loss in the approximate solutions. In the polygonal scheme, we divide the (x, t)-plane into finite districts, on each of which the forward (resp. backward) waves are of the same type: forward (resp. backward) rarefaction or compression.
Because the density increases when it crosses a compressive wave, it seems that only districts including forward and backward rarefaction waves directly make the density decreasing. However, things are more complicated than they appear. In fact, for rarefaction-rarefaction districts adjacent to initial line, we can apply directly a similar argument used in [17] to obtain the desired lower bound on density. However, the major difficulty that we have to conquer in this paper is how to analyze those rarefaction-rarefaction districts far away from the initial line. These rarefaction waves have passed some compressions in the opposite families before reaching the rarefaction-rarefaction districts. So max x =y s(x, t) − s(y, t) x − y and max x =y r(x, t) − r(y, t) x − y , might increase as time evolves. In order to obtain a sharp lower bound on density, we need to carefully analyze all three types of districts: rarefaction-rarefaction, rarefaction-compression and compression-compression districts. Unlike many previous analysis, where the changes of density are monitored across the boundaries of districts, or jump edges, we track the wave packs in each district, for which we introduced the concepts of rarefactive and/or compressive characters in Section 3 below. With the help of these characters, we are able to track the variation of density precisely. Furthermore, in Definition 4.1, we introduced an interesting function a(t) which measures the possible density loss. Using one of our key new ideas, we are able to prove in Lemma 4.3 that a(t) is not increasing in time, in the scheme approximating general smooth solutions. Using this lemma, we could sew up the density estimates obtained in each district into a global one up to time T in the regime of Lipschitz continuous solutions.
This paper is organized as follows. In Section 2, we review the explicit example given in [9] where density decays to zero in time with an order of O(1 + t) −1 . In Section 3, we review the polygonal scheme and define Rarefaction/Compression character. In Section 4, we prove the main theorem on lower bound of density. Finally, as an application, in Section 5, we use the time-dependent lower bound (5) obtained in Theorem 1.2 to improve the life-span estimates for classical soluitons of (1) given in [7].
2. Exact interaction between two centered rarefactions. In this section, we review a classical example for interaction between two centered rarefaction waves provided in Section 82 of [9], where detail calculations are given. To be consistent with classical literatures, (only) throughtout this section, instead of s and r, we use the following slightly different Riemann invariants S and R, In fact, S (resp. R) is different from s (resp. r) only by some constants. The following relation is crucial to measure density, For simplicity, in Figure 1 below, we only consider an interaction between two centered rarefaction waves, where we assume that the first interaction happens at t =t > 0 and x = 0, with constant state (S,R) below the point (0,t). Furthermore, we assume that u(0,t) = 0, henceS Again, for simplicity, we assume that S in the left state of interaction is always positive and R in the right state of interaction is always negative.
Then using the fact that S and R are constant along forward and backward characteristics, respectively, in [20], Riemann first found the following equation: . where t(S, R) is the time when the interaction ends, F (z 1 , z 2 , z 3 , z 4 ) is a hypergeometric function and .
When σ is a positive integer N or equivalently where P σ−1 (z) is the Legendre's function, which is a (σ − 1)-th order polynomial.
Recall that we assume that S in the left state of interaction is always positive and R in the right state of interaction is always negative. Let |S| and |R| be both very small, then density ρ at the point where interaction ends (at time t(S, R)) is very close to zero. Using (7), to the leading order, we have where we used (6), (9)-(10), P σ−1 (z) is a (σ − 1)-th order polynomial and 1 S (S 2 − SR) >S is uniformly positive. When |S| and |R| both approach zero, It is clear that the initial density in this explicit example is uniformly positive, as illustrated in the right picture of Figure 1.
For general γ > 1, similar calculation on the density lower bound for the interaction of centered rarefactions can be found in [14].
3.1. Pressure, Riemann invariants and standard states. We first give the polygonal approximation of p(v) as follows: for any given positive integer n, let v (n) 0 = 1, and v (n) k (for integer k) be determined by the recurrence formula It is easy to check that for each fixed n there exists a unique sequence v , defined by (11), such that By connecting the points v In the second step, based on the definition of p (n) (v), we give the polygonal approximation of Riemann invariants r and s. Denoting One quickly has where k is an integer and n is a positive integer. We define which are corresponding to the Riemann invariants r and s defined in (3), respectively.
Finally, we give the definition of standard states: where i and j are integers satisfying For convenience, we might omit the superscript (n) if there are no confusions.

Riemann problems.
Based on the definitions introduced in the previous subsection, now we consider the following Riemann problem: with j+M −N , where M (and N ) can be −1, 0 or 1. Clearly, we have The solution of the Riemann problem (14) consists of |M | + |N | + 1 standard states, divided by |M | + |N | jump discontinuities (straight lines centered at the origin).
To calculate the middle state (r Then it is easy to have Remark 1. We note that • if M = 1 (resp. N = 1), the backward (resp. forward) jump discontinuity describes a rarefactive wave. • If M = 0 (resp. N = 0), there are no backward (resp. forward) jump discontinuity. • if M = −1 (resp. N = −1), the backward (resp. forward) jump discontinuity describes a compressive wave. The definitions of rarefaction and compression are in Definition 1.1. We refer the readers to [2,5] for more details on wave curves for rarefaction and compression waves.
When M is −1 or 1, from (11), the slope of the backward jump is

GENG CHEN, RONGHUA PAN AND SHENGGUO ZHU
When N is −1 or 1, from (11), the slope of the forward jump is 3.3. The polygonal scheme and Rarefactive/Compressive characters. For any given Lipschitz continuous initial data (r 0 , s 0 ) with u 0 and v 0 uniformly bounded and v 0 uniformly away from zero, similar to [17], we can find a sequence of piecewise constant functions (u = +∞, such that for every integer 1 ≤ α ≤ j, uniformly, and Here, in fact, we can choose the approximation piecewise constant profiles of s (n) (x) and r (n) (x) such that when x is varying from x α to x α+1 , the variations of s and r values are equal to ± 2 n , which more precisely means, s (n) (x α+1 + ε) − s (n) (x α + ε) = ± 2 n and r (n) (x α+1 + ε) − r (n) (x α + ε) = ± 2 n when ε is small enough. So as n → ∞, (18) is satisfied.
For any positive integer n, we solve the Riemann problem (14) at each discontinuity. Note the left, right and middle sates in the solution of each Riemann problem (14) are still standard states, split by jump discontinuities. Let these jump discontinuities evolve. When two jumps in different characteristic families interact with each other, new Riemann problem appears, which can also be solved according to the method established in the previous subsection and there are exactly two outgoing jumps. Before the possible interactions between jumps of the same family, we get a well-defined polygonal scheme, including finitely many jumps, because the total number of interactions are finite and the number of outgoing discontinuities equals to the number of incoming discontinuities in one interaction.
In [19], the authors show that under some regularity condition on the initial data (u 0 , v 0 ), the approximation solutions (u (n) , v (n) ) in the polygonal scheme used in current paper are well-defined, i.e. there is no interaction between jumps of the same family, in a time interval t ∈ [0, T ] with T > 0 only dependent on the C 1norm of (u 0 , v 0 ) but independent of n. Furthermore, the approximation solutions converge to a Lipschitz continuous solution for p-system when n → ∞. We will give more details on this local-in-time convergence result later.
In this subsection, we first assume that there is no interaction between jumps of the same family.
Definition 3.1. In the polygonal scheme, for any positive integer n, the (x, t)-plane is divided into finite blocks by finitely many jump discontinuities. If a block is a diamond, we call it a diamond block or diamond. Each jump discontinuity may also be divided into finite many pieces, which are denoted by jump edges, by finitely many intersection points with other jumps.
To be precise on the definitions of a block or a jump edge, we note s (n) and r (n) are both constants inside each block and on each side of a jump edge, or in another word, there are no other jump discontinuities going inside a block or a jump edge.
To obtain a good lower bound on density, it is crucial to study the variation of a jump edge inside a characteristic tube, such as the propagation of edge l 1 in the forward characteristic direction in Figure 2.
First, we define the Rarefactive/Compressive (R/C) character on a jump edge.
Definition 3.2 (R/C character on a jump edge). We classify the jump edges into four types: R r , R c , C r and C c . Most of time, we add an arrow to denote forward or backward character, respectively. More precisely, a backward (resp. forward) jump edge l 1 (resp. l 2 ) in the scheme, shown in Figure 2, is said to be: ii. R c (resp. R c ), if u − < u + and u + > u ++ (resp. u − < u + and u −− > u − ). iii. C r (resp. C r ), if u − > u + and u + < u ++ (resp. u − > u + and u −− < u − ). iv. C c (resp. C c ), if u − > u + > u ++ (resp. u −− > u − > u + ).
For the left-most forward (resp. right-most backward) jump edge in the scheme which is unbounded from its left (resp. right) hand side, we always say this jump edge belongs to either C c or R r by checking the relation of u from the right (resp. left) boundary following the table above.
For simplicity, we always use C to denote C r or C c character. Figure 2. Definition of forward (resp. backward) R/C characters for the jump edge l 1 (resp. l 2 ). The picture is on the (x, t)-plane.

Remark 2.
In the definition of R r , R c , C r and C c , the capital letter denotes the character on the boundary behind the edge (u increases from left to right for a rarefaction boundary) and the subscript denotes character on the boundary ahead of the edge.
In Figure 2, we use the subscripts a and b to denote states ahead of and behind a jump wave front. By (15), one always has v b > v a for a R r or R c jump edge; and v b < v a for a C r or C c jump edge. In the R c and C r pieces, v aa = v b .
Then we define the R/C character on any blocks. Definition 3.3 (R/C character in a block). A block is called a R r R r block if its South-West and South-East boundaries are R r and R r , respectively.
Similar definitions are also for R r R c R c R r , R c R c , C R r , C R c , R r C, R c C and C C blocks. Lemma 3.4. The forward (resp. backward) jump edges l SW and l N E (resp. l SE and l N W ) shown in Figure 3 belong to the same type.
Proof. By (13) and (15), we have . Summing up these equations, one obtains with states 1∼4 given in Figure 3, hence the monotonicity of u is preserved in both forward and backward directions, which is enough to prove this lemma by Definition 3.2.
Definition 3.5 (R/C character in a district). Given a block in certain type (for example a R r R r block), we define a district in that type (for example a R r R r district) to be the largest connected set which includes the given block and consists of only blocks in the same type.
Finally, we prove that v decays in some direction, when we restrict our consideration on districts which are not in R r R r type. Lemma 3.6. If the forward character of a district D is C or R c , then v on any block adjacent to and above North-West boundary of D is not larger than the maximum v values on blocks adjacent and below South-East boundary of D.
And all Φ (n) values on blocks in D are at most 1/n larger than the maximum Φ (n) value on blocks adjacent and below South-East boundary of D.
Symmetric decay of v happens if the backward character of a district D is C or R c .
The proof of this lemma is obvious from Remark 2. This lemma is corresponding to the fact that density increases from behind to ahead of a compression simple wave.
4. The lower bound on density in the scheme. This section will be devoted to prove density's lower bound estimate shown in Theorem 1.2. The key idea in proving (5) is to define a function a (n) (t) for any n, which is not decreasing on t. The monotonicity of a (n) (t) will finally lead to a lower bound on density, under the help of the local convergence theorem for the polygonal scheme in [19]. In this section, we always assume that jumps in the same family do not interact, where this assumption will be removed in the end of this section.
To define a (n) (0), in the first step, we modify some blocks adjacent to the initial line t = 0 to diamonds, as in Figure 4 and the left picture of Figure 5. After these modifications, we call all interior and boundary diamonds as complete diamonds, also shown in Figure 4.     For t > 0, it is not necessary to modify the diamond again. We thus define a (n) (t) in a similar way. We first give a lemma on the initial data.
then the density ρ (n) x, L and M 0 is a positive constant depending on the uniform upper and lower bound on initial density but independent of n. Proof. The first claim is clearly true because each state below the curve t = L 0 (x) is an initial state in the scheme.
To prove (21), it is enough to show that for any jump edge P Q with endpoints P and Q on the curve t = L (n) where ε n goes to zero as n goes to infinity. We divide jump edges on the curve t = L (n) 0 (x) into two types: jump edge intersecting with the initial line and jump edge not intersecting with the initial line. Without loss of generality, we only consider two forward jump edges l 1 and l 2 in Figure 6. We denote three line segments on t = 0 divided by adjacent backward jumps as B j−1 , B j and B j+1 , respectively, which are shown in Figure 6.
Noticing that l 1 is R r , from (15), Definition 3.2 and the discussion in Subsection 3.1, we have r = 2(k−1) n , 2k n and 2(k+1) n in backward tubes, which means that all connected blocks are divided by two adjacent backward jumps, including B j−1 , B j and B j+1 , respectively, where k = 1 2 (i + j) for some integers i and j. By studying the shaded triangle in Figure 6, it is easy to get that there exists a constant M only depending on the upper and lower bounds of initial density, such that M ≥ |B j | ∆ l1 x as n large enough, where ∆ l1 x means the length of projection of l 1 onto the line t = 0. Then we have Hence (22) is clearly correct, because the right hand side of (23) is bounded above by J as n goes to infinity, where we have also used (18). For l 2 , noting that ∆ l2 x ≥ ∆ l 2 x where l 2 is parallel to l 2 , because l 2 is R i , we know that two backward jumps enclosing l 2 open up. A more detailed argument on this fact can be found in Step 2.1 in the proof of Lemma 4.3. Now we change the problem to a problem for l 2 which intersects with t = 0, hence similar as the case for l 1 , we can prove (22).
This completes the proof of the lemma.
The following lemma plays a key role in this paper. Proof. We divide our proof into two steps. For convenience, we omit the subscript (n) in the proof of this lemma.
Step 1. Suppose jump edges P u P l and P u P l are two lower boundaries of a diamond Ω, where P u , P u and P l are three vertexes of Ω, and specially P l is the lowest vertex.
Then we prove that: for any time T ≥ 0, P u P l and P u P l are either both in or both not in the collection of selected diamonds at time T .
Actually, if P l is in the region t ≥ T , then Ω is a selected diamond. If P l is not in the region t ≥ T , then diamonds containing P u P l or P u P l as north boundary are not selected. Thus except these diamonds, the only diamond including P u P l or P u P l is Ω. Hence P u P l and P u P l are either both selected or both not selected.
Step 2. Recall that the R/C characteristic does not change along forward and backward characteristic directions. Also using the claim proved in Step 1, to prove the lemma, we only have to show that, in any diamond, the minimum difference in x for any R r and R r south jump edges is less or equal to the minimum difference in x for any R r and R r north jump edges.
We discuss case by case for diamonds including R r or R r .
Case (2.1.) We consider a R r C or a R r R c diamond shown in Figure 7. We use subscripts E, W , N and S to denote functions related to east, west, north and south endpoints of the diamond, respectively. And we use subscripts N E, N W , SE and SW to denote functions related to North-East, North-West, South-East and South-West boundary jump edges of the diamond, respectively. Especially we use l N E , l N W , l SE and l SW to denote the North-East, North-West, South-East and South-West boundary jump edges of the diamond. We want to show that . From left to right: the middle diamonds are: R r C c ; R r C r ; and R r R c diamonds, respectively, where forward character always goes first. The picture (x, t)-plane.
By studying the three possible cases in Figure 7 using (15)∼(16), we always have where recall that subscripts N , S, W , E denote the north, south, west and east endpoints of the middle diamond in the figure, respectively. Then we could give an easy geometric proof for (24) in Figure 8. In fact, drawing two dash lines parallel to l N W and l SW , respectively, then by (25), we know the parallelogram is inside the diamond, although in R r C r and R r R c diamonds in Case (2.2.) However, the easy geometric proof in the previous part fails for R r R r interaction where the second inequality in (25) is in the opposite direction. Instead we use the similar idea as the one used in [17] to cope with a R r R r diamond.  Figure 9. The center diamond is a R r R r block. The picture is on the (x, t)-plane.
We use B and D to denote the difference in x and t for each adjacent pair of endpoints on the diamond, respectively, such as Then clearly one has B N W + B N E = B SW + B SE , Denote that it is easy to deduce that