The nonlinear diffusion equation of the ideal barotropic gas through a porous medium

Abstract The nonlinear diffusion equation of the ideal barotropic gas through a porous medium is considered. If the diffusion coefficient is degenerate on the boundary, then the solutions may be controlled by the initial value completely, the well-posedness of the solutions may be obtained without any boundary condition.


Introduction
Consider the motion of the ideal barotropic gas through a porous medium, let , V and p be the gas density, the velocity and the pressure respectively. Then, Antontsev-Shmarev [1] pointed out the motion is governed by the mass conservation law t C div. V / D 0; the Darcy law V D k.x/rp; k.x/ is a given matrix and the equation of stage p D P . /. It is usually assumed that P .s/ D s˛with ;˛Dconst. The above conditions then lead to the semi-linear parabolic equation for the density C˛d iv.k.x/r 1C˛/ : If we additionally assume that p may explicitly depend on .x; t / and has the form p D .x;t/ , the equation for becomes t D div.k.x/ r .x;t/ / and can be written in the form t D div.k.x/ r C . log /k.x/ r /; .x; t / 2 Q T D .0; T /; where is a bounded domain in R N with appropriately smooth boundary. Antontsev-Shmarev [1] made a simplified version of equation (1), u t div.juj .x;t/ ru/ D f .x; t /; considered the usual initial boundary value conditions u.x; 0/ D u 0 .x/; x 2 : u.x; t / D 0; .x; t / 2 @ .0; T /: (4) and gave the following and for 0 Ä t 1 Ä t 2 Ä T , it holds Let .x; t / be a appropriately smooth function in Q T . If Antontsev-Shmarev [1] had proved that the problem (2)-(3)-(4) has at least one weak solution in the sense of Definition 1.1. The solution is bounded and satisfies the estimate kuk 1;Q T Ä K.T / with the constant K.T / from condition (7). In the present work, we limit ourselves to the study of the following equation where .x/ D dist.x; @ /,˛> 0. If˛D 0 and .x; t / Á m D const, equation (8) is the usual porous medium equation Also, when m D 1, f D 0, equation (9) is regarded as a nonlinear heat equation which has been studied in many well-known monographs or textbooks, for examples one can refer to [2][3][4][5][6][7], where a wide spectrum of methods is used. The function k.u/ has the meaning of nonlinear thermal conductivity, which depends on the temperature u D u.x; t /. Meanwhile, if 0 > m > 1, equation (9) is called a fast diffusion equation. The name "fast diffusion" is related to the fact that since the heat conductivity is unbounded in the unperturbed (zero temperature) background, the heat propagates from warm regions into cold ones much faster than it propagates in the case of constant (m D 0 in (9)) heat conductivity, and even faster than in the case m > 0, in which the speed of propagation of perturbations is finite. Different from equation (9), equation (8) reflects that the diffusion process depends on the distance function .x/ from the boundary. In particular, ˛D 0 on the boundary means that the equation is degenerate on the boundary. If we want to consider the initial-boundary value problem of equation (8) with .x; t / Á D const > 1, the initial value (3) is always necessary. But, the boundary value condition (4) may be superfluous. To see that, we consider the equation @u @t D @ @x i . ˛. x/a.u/ @u @x i /; where D const , A.u/ D u , a.u/ D u 1 . For small Á > 0, let Obviously h Á .s/ 2 C.R/, and h Á .s/ 0; j sh Á .s/ jÄ 1; j S Á .s/ jÄ 1I lim If u and v are two classical solutions of equation (11) with the initial values u 0 ; v 0 respectively, then we have where n D fn i g is the inner unit normal vector of . Let Á ! 0. Then we have It means that the classical solutions (if there are) of equation (11) are completely determined by the initial value, in other words, the solutions are free from the limitation of the boundary condition. The phenomena that the solution of a degenerate parabolic equation may be free from the limitation of the boundary condition also can be found in [8][9][10] et.al.
In this paper, we will study the well-posedness of the solutions to equation (8) with the initial value (3) but without any boundary condition. When we study the stability, we encounter two obstacles. The first one is that the solution lacks the regularity on the boundary, the second one is how to deal with the nonlinearity of juj .x;t/ .
We denote that and suppose that u 0 2 L 1 . /; Definition 1.2. A function u.x; t / is said to be the weak solution of equation (8) with the initial value (3), if The main results in our paper are the following theorems.
Roughly speaking, we may conjecture only if˛> 0, the stability (19) may be true without the condition (18) or (20). Theorem 1.4 and Theorem 1.5 have verified the fact partly. Recently, the author [12] had studied the equation with˛> 0, and had shown that the uniqueness of the solutions of equation (20) is true without any boundary value condition. Meanwhile, Ji L r K i Benedikt et.al [13,14] had studied the equation with 0 <˛< 1, and shown that the uniqueness of the solutions of equation (22) is not true. From the short comment, one can see that the degeneracy of the coefficient ˛p lays an important role in the well-posedness of the solutions, it even can eliminate the action from the source term f .u; x; t / . The paper is arranged as follows. In the first section, we give a brief introduction and narrate the main results. In the second section, we prove the existence. In the third section, we prove the stability of the solutions.

The existence
Proof of Theorem 1.3. Let u 0 2 L 1 . / satisfy (14). Consider the regularized problem of equation (8) u t div.a."; u; x; t /ru/ D f .x; t /; .x; t / 2 Q T ; with the initial-boundary value condition (3)-(4). Here a."; u; x; t / D . ˛C "/." C ju " j/ .x;t/ . Then 0 < C 0 ."/ Ä a."; u; x; t / Ä C."/; Similar to [1], by Schauder Fixed Point Theorem, we know that the regularized problem has a solution u " in the sense of Definition 1. Simplifying and then integrating this relation in t, we obtain the following estimates for the solutions of equation (23) ku " . ; t /k 2k; Ä t Z 0 kf k 2k; dt C ku 0 k 2k; ; Passing to the limit when k ! 1, we have By multiplying (23) by u " , we are easily to obtain that " Q T a."; u; x; t /jru " j 2 dxdt Ä c: If the constant C C 1, then By multiplying (23) by v 2 H 1 0 . /; kvk H 1 0 D 1, we easily to obtain that j < u "t ; v > j Ä " Q T a."; u " ; x; t /jru " j 2 dxdt C c Ä c; which implies that ku "t k L 2 .0;T IH 1 . // Ä c: The uniform estimates (27)-(28), using the result of [11,Sec.8], yield relative compactness of the sequence fu " g in L s .Q T / with some s 2 .1; 1/. Then we can choose a subsequence fu l g Â fu " g such that fu l g is compact in L s .Q T /. By the arbitrary of , we have u l ! u; a.e. i n Q T : Moreover, by (25)-(28), when l ! 1, we have u l * u; weakly star in L 1 .Q T /; u lt * u t 2 L 2 .0; T I H 1 . //; p a.l; u; x; t /u lx i * i ; weakly in L 2 .Q T /; where a.l; u; x; t/ D a."; u; x; t / j u l Du " , D f i W 1 Ä i Ä N g and every i is a function in L 2 .Q T /. In order to prove the theorem, we firstly prove D q ˛j uj .x;t/ ru; in L 2 .Q T /: For any '.x; t / 2 C 1 0 .Q T /, noticing that u l ! u, a.e. in Q T , then Here, we have used the fact jr j D 1, and Secondly, we prove (17). For any given small > 0, large enough k; l, we declare that where c .t / is independent of k; l, and lim t !0 c .t / D 0. By (16), for any ' 2 C 1 0 .Q T /, Supposing that .x/ 2 C 1 0 . / such that ju k j ru k ju l j ru l r.u k u l / S 0 Á .u k u l /dxd D 0: ju k j ju l j /ru l r.u k u l / S 0 Á .u k u l /dxd By (13), lim Á!0 sS 0 Á .s/ D 0, and using the fact By (36), (37), we have which means (33) is true. Now, for any given small r, if k; l are large enough, by (33), we have letting ! 0, we get (17). Theorem 1.3 is proved.

The stability
Proof of Theorem 1.4. For a small positive constant > 0, let Now, if u 0 and v 0 only satisfy (14), let u; v be two solutions of equation (8) with the initial values u 0 ; v 0 respectively.
We now calculate the terms of (47) as follows.