Iwabuchi, T. and Ogawa, T. Osaka J. Math. 53 (216), 919 939 ILL-POSEDNESS ISSUE FOR THE DRIFT DIFFUSION SYSTEM IN THE HOMOGENEOUS BESOV SPACES TSUKASA IWABUCHI and TAKAYOSHI OGAWA (Received February 13, 215, revised September 11, 215) Abstract We consider the ill-posedness issue for the drift-diffusion system of bipolar type by showing that the continuous dependence on initial data does not hold generally in the scaling invariant Besov spaces. The scaling invariant Besov spaces are ÈB 2np p, (Ê n ) with 1 p, ½ and we show the optimality of the case p 2n to obtain the well-posedness and the ill-posedness for the drift-diffusion system of bipolar type. 1. Introduction We consider the ill-posedness issue for the initial value problems of a drift-diffusion equation of bipolar type: (1.1) t u ½u Ö (uö), t, x ¾ Ê n, t Ú ½Ú Ö (ÚÖ), t, x ¾ Ê n, ½ Ú u, t, x ¾ Ê n, u(, x) u (x), Ú(, x) Ú (x), x ¾ Ê n, where u and Ú are the particle density of negative and positive electric charge, is the coupling constant and we assume 1. u and Ú are given initial data. The system (1.1) was originally considered for an initial boundary value problem with Dirichlet or Neumann boundary condition as simplest model of a semi-conductor device simulation and we refer to [1, 6, 8, 11, 19, 24] for the related results. As the model of the semiconductor device simulation, the mono-polar model is also considered; (1.2) t u ½u Ö (uö), t,x ¾ Ê n, ½ u, t, x ¾ Ê n, u(, x) u (x), x ¾ Ê n. 21 Mathematics Subject Classification. Primary 35K55; Secondary 35K8.
92 T. IWABUCHI AND T. OGAWA The well-posedness issue was considered in both models (1.1) and (1.2) (see for instance, [9], [14], [15], [17], [18], [22], [3]). We note that the mono-polar model is considered as the limiting model of the Keller Segel system in the chemotaxis and there are large literatures for this direction [3], [5], [9], [12], [14], [15], [2], [21], [22], [28]. The both of the problems (1.1) and (1.2) share the common scaling invariant property. Namely under the scaling transform (1.3) u (t, x) 2 u( 2 t, x), Ú (t, x) 2 Ú( 2 t, x), (t, x) ( 2 t, x) with, the equations in (1.1) and (1.2) remain invariant. Then the common invariant space by the scaling (1.3) in the Bochner space L (Ê Á L p (Ê n )) is given by a restriction on (, p) with 2 2 n p. In particular for ½, the solution is consistent and we reach the invariant function spaces for (1.1) as (u, Ú, ) ¾ L n2 (Ê n ) L n2 (Ê n ) L ½ (Ê n ). Such a critical space can be generalized by term of the homogeneous Besov spaces with negative regularity indices such as B È 2np p, (Ê n ). According to the analytical and scaling structure of the nonlinear coupling term, the most of basic feature for the solutions to both the bipolar system (1.1) and the mono-polar system (1.2) are similar and common except the limiting class of the well-posedness. Namely, there appears a difference between (1.1) and (1.2) for the invariant limiting function spaces with low regularity. To specify the critical space for the well-posedness precisely, we necessarily introduce the scaling invariant Besov spaces B È 2np p,½ (Ê n ) with 1 p ½. It is shown by Iwabuchi [9] that the initial value problem (1.2) of the monopolar type is well-posed for small initial data in the Besov spaces B È 2np p,½ (Ê n ) with n2 p ½ and ill-posed in B È 2 ½,½ (Ên ). This shows that the case p ½ is threshold for the wellposedness issue of the mono-polar type (1.2). On the other hand for the equation (1.1), Zhang Liu Ciu [3] showed that the problem is well-posed in B È 2np p,½ (Ê n ) with n2 p 2n, however no ill-posed result can be found in the literatures. In this paper we show that the critical space for the well-posedness and the illposedness to the equation (1.1) is identified as p 2n through the study of the illposedness in the Besov spaces B È 2np p,½ (Ê n ) (2n p ½). We define the homogeneous Sobolev spaces and the Besov spaces and state our theorems. We denote the function spaces of rapidly decreasing functions by S(Ê n ), tempered distributions by S ¼ (Ê n ), and polynomials by P(Ê n ).
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 921 DEFINITION (the homogeneous Sobolev spaces). homogeneous Sobolev space H È s p (Ê n ) is defined by For any s ¾ Ê, 1 p ½, the ÈH p s H È s p (Ê n ) Ï { f ¾ S ¼ (Ê n )P(Ê n ) f ÈH s p Ï F 1 [ s Ç f ()]L p (Ê n ) ½}, where F 1 denotes the inverse Fourier transform. DEFINITION (the homogeneous Besov spaces). following: supp Ç { ¾ Ê n 2 1 2}, j¾ Let ¾ S(Ê n ) be satisfying the Ç 2 j 1 for ¾ Ê n Ò {}, where Ç is the Fourier transform of, and let { j } j¾ be defined by j (x) Ï 2 nj (2 j x) for j ¾, x ¾ Ê n. Then, for any s ¾ Ê, 1 p, ½, the homogeneous Besov space È B s p, (Ê n ) is defined by ÈB s p, (Ên ) Ï { f ¾ S ¼ (Ê n )P(Ê n ) f ÈB s p, Ï {2 s j j f L p (Ê n )} j¾ l () ½}. REMARK. One can regard the above homogeneous spaces as a subspace of S ¼ (Ê n ) for some s, p, q. Indeed, if s and p satisfy s np, then the homogeneous Sobolev space È H s p (Ê n ) is equivalent to u ¾ S ¼ (Ê n ) u È H s p ½, u j¾ j u in S ¼ (Ê n ) µ. If s np with 1 q ½, or s np with q 1, the Besov space È B s p,q (Ê n ) is also equivalent to u ¾ S ¼ (Ê n ) u È B s p,q ½, u j¾ j u in S ¼ (Ê n ) These equivalence are due to the argument by Kozono Yamazaki [16]. µ. Theorem 1.1. Let n 2, 1 and let p, satisfy (1.4) 2n p ½ and 1 ½, or p 2n and 2 ½.
922 T. IWABUCHI AND T. OGAWA Then, there exist a sequence of times {T N } N with T N (N ½) and a sequence of smooth and rapidly decreasing initial data {u, N } N, {Ú, N } N (N 1,2,) such that the corresponding sequence of smooth solutions {u N } N, {Ú N } N to (1.1) with u N () u, N and Ú N () Ú, N satisfies lim N½ u, N ÈB 2np p,, lim N½ Ú, N ÈB 2np p,, lim N½ u N (T N ) ÈB 2np p, ½, lim N½ Ú N (T N ) ÈB 2np p, ½. REMARK. We should make the function space clear where the solution in the above theorem belongs to. There is no result of well-posedness to the system (1.1) in Besov spaces B È s p, (Ê n ) under the condition (1.4). In the proof of theorem, we justify the solutions in the space C([, T ], M n,1 (Ê n )), where M n,1 (Ê n ) is the modulation space specified in Section 2. We note that these solutions are in the space C([, T ], B È 1 n,1 (Ên )), and it is known that the local well-posedness is obtained in the space B È 1 n,1 (Ên ) by the result [3]. Therefore initial data and corresponding solutions can be constructed with enough smoothness to guarantee justification of solutions. REMARK. We note that when n 2, p 4 and 2, the Besov space B È 32 4,2 (Ê 2 ) is the critical space for the well-posedness and ill-posedness. Indeed, one can show the global well-posedness for small initial data in B È 32 4,2 (Ê 2 ). In general, for p 2n with 1 2 except for (n, ) (2, 2), one can show the ill-posedness in the analogous way in Theorem 1.1. The main reason why the limitation of the well-posedness class is different in two problems (1.1) and (1.2) is because it depends on how much order the nonlinearity can exhibit derivatives and hence it depends on the symmetry of the nonlinear coupling. For the equation (1.2) of mono-polar type, the nonlinear term uö( ½) 1 u satisfies x j u x j ( ½) 1 u 1 2 x j ( ½){(( ½) 1 u)( x j ( ½) 1 u)} (1.5) x j Ö {(( ½) 1 u)(ö x j ( ½) 1 u)} 1 2 2 x j {(( ½) 1 u)u}, which was observed in [9] and hence we can treat the nonlinear term Ö (uö( ½) 1 u) as a doubly divergence form such as Ö 2 {(Ö 2 u)u}. This enables us to treat the equation in the weaker Besov space up to B È 2np p,½ with 2 p ½ and B È 2 ½,2. On the other hand for the bipolar type (1.1), the nonlinear term Ö (uö( ½) 1 Ú) has lack of symmetry in the nonlinear structure which prevents to have such a good expression as (1.5).
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 923 To be more precise, let be a characteristic function which support is [ 1, 1] and e 1 Ï (1,,, ), and we take initial data as u N 2 np F 1 [( Ne 1 )], Ú N 2 np F 1 [( Ne 1 )]. We note that the Fourier transforms of u and Ú are supported locally at particular frequency N and N, respectively, and u ÈH 2np, Ú p ÈH 2np are independent of N, p where H È s p (Ê n ) is the homogeneous Sobolev space. By the Duhamel formula, we write the solution by the integral equations: (1.6) u(t) e t½ u Ú(t) e t½ Ú t t e (t )½ Ö (uö( ½) 1 (Ú u)) d, e (t )½ Ö (ÚÖ( ½) 1 (Ú u)) d. Since both of u and Ú are treated similarly, we consider the nonlinear term of u only. Then, we approximate the nonlinear part in the right hand side by a linear solution u e ½ u, Ú e ½ Ú and it follows that (1.7) t e (t )½ Ö (e ½ u Ö( ½) 1 e t½ (Ú u )) d ÈH 2np p F 1 2np F 1 Ê n Çu ( )(ÇÚ () Çu ()) t 2 2np F 1 2np Ê n e (t )2 e 2 e 2 d d L p L p Çu ( )(ÇÚ () Çu ()) e t2 (e 2t( ) 1) 2 d 2( ) Ê n Çu ( )Ú () N N 2 1 2N 2 d L p F 1 [ 2np ()] L p N 2(2 np) N 1 N 2 N 1 2np. The last term diverges as N ½ if p 2n. On the other hand for the part with the convolution of Çu and Çu, we have from the structure Ö (uö( ½) 1 u) Ö 2 {(Ö 2 u)u}
924 T. IWABUCHI AND T. OGAWA by (1.5) and the support of Çu Çu being in the neighborhood of the frequency 2N (1.8) F 1 2np Ê n Çu ( )Çu () e t2 (e 2t( ) 1) 2 d 2( ) F 1 [ np ( 2N)] L p N 2(2 np) N 2 N 2 N np L p and the last term is bounded for all p 1 and N ¾ Æ. Therefore, one can expect the divergence of the nonlinear term in (1.6) when p 2n as N ½ by (1.7) and (1.8), while the norms of the initial data are bounded by some constant independent of N. For the precise proof, we introduce an asymptotic expansion of solutions in terms of the order of the iterative succession by the initial data, and we justify it in the modulation space M n,1 (Ê n ) which is shown in Section 3. The usage of the modulation space is useful whenever we consider the nonlinear term since M n,1 (Ê n ) is a Banach algebra. Then we show the term observed in (1.7) tends to ½ and the other terms in the asymptotic expansion are small. In Section 4, we give a sequence of initial data and solutions satisfying the statement of our theorem by the use of the asymptotic expansion introduced in Section 3. We treat the system (1.1) with 1 only in the following sections since the case 1 can be treated analogously. Finally in Section 5, we compare the results on the local well-posedness for the incompressible Navier Stokes system in three dimensions. 2. Preliminary We introduce the modulation spaces M p, (Ê n ) and show some facts for the bilinear term of functions in the modulation spaces which will be used in the proof of our theorem. In what follows, we denote various constants simply by C. DEFINITION (the modulation spaces). satisfies Let k is the Fourier window function that supp Ç k { ¾ Ê n k j 1 j k j 1 for j 1, 2,, n}, k¾ n Ç k () 1. Then, for any 1 p, ½, the modulation space M p, M p, (Ê n ) is defined by M p, (Ê n ) Ï { f ¾ S ¼ (Ê n ) f Mp, Ï { k f L p (Ê n )} k¾ n l ( n ) ½}. Lemma 2.1 ([7], [25], [27]). (i) M p1, 1 (Ê n ) M p2, 2 (Ê n ) if p 1 p 2 and 1 2. (ii) Let 1 p, p ¼ ½ satisfy 1p 1p ¼ 1. Then, it holds that M p,min{p, p ¼ }(Ê n ) L p (Ê n ) M p,max{p, p ¼ }(Ê n ).
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 925 (iii) Let 1 p, ½. Then, there exists C such that (2.1) Öe t½ f Mp, Ct 12 f Mp,. (iv) Let 1 p ½. Then, there exists C such that (2.2) f g Mp,1 C f Mp,1 g Mp,1. See for the proof, [7], [25], [27]. The following is the lemma for the estimate in Besov spaces. Lemma 2.2. (i) [13] Let s s 1 and 1 p ½. Then, there exists C such that (2.3) e t½ f ÈB s 1 p,1 Ct (s 1 s )2 f ÈB s p,½, for all f ¾ B È s 1 p,½ (Ên ). (ii) [9] Let p, p 1, p 2 satisfy 1 p, p 1, p 2 ½, 1p 1p 1 1p 2. Then, there exists C such that (2.4) f Ö( ½) 1 g gö( ½) 1 f ÈB 1 p,½ C f È B 1 p 1,1 g È B 1 p 2,1, for all f ¾ È B 1 p 1,1 (Ên ) and g ¾ È B 1 p 2,1 (Ên ). 3. Asymptotic expansion We introduce the asymptotic expansion of the solution to (1.1) by some small parameter as u U U 1 2 U 2 3 U 3, Ú V V 1 2 V 2 3 V 3, with the initial data u ³ ³ ³ 1 2 ³ 2, Ú 1 2 2. For simplicity, let F be defined by F(u, Ú) Ï Ö (uö( ½) 1 ÖÚ),
926 T. IWABUCHI AND T. OGAWA and let us consider 1 of the equation (1.1). If u, Ú satisfy the equation (1.1), we have the followings on the term of order k with k, 1, 2, Ï 1 Ï 2 Ï ( t ½)U F(U, V U ), ( t ½)V F(V, U V ), U 1 () ³, V 1 (), ( t ½)U 1 F(U, V 1 U 1 ) F(U 1, V U ), ( t ½)V 1 F(V, U 1 V 1 ) F(V 1, U V ), U 1 () ³ 1, V 1 () 1, ( t ½)U 2 F(U, V 2 U 2 ) F(U 1, V 1 U 1 ) F(U 2, V U ), ( t ½)V 2 F(V, U 2 V 2 ) F(V 1, U 1 V 1 ) F(V 2, U V ), U 2 () ³ 2, V 2 () 2, respectively. For our proof of the theorem, let ³ k and k for k,2,3,4,5, without k 1 and we consider the initial data u ³ 1, Ú 1. The term of order k can be reduced as follows: Ï 1 Ï 2 Ï k Ï ( t ½)U, ( t ½)V, U 1 (), V 1 (), ( t ½)U 1, ( t ½)V 1, U 1 () ³ 1, V 1 () 1, ( t ½)U 2 F(U 1, V 1 U 1 ), ( t ½)V 2 F(V 1, U 1 V 1 ), U 2 (), V 2 (), ( t ½)U k ( t ½)V k k 1 k, k 1, 1 k 1 k, k 1, 1 U k (), V k (), F(U k1, V k2 U k2 ), F(V k1, U k2 V k2 ), for k 2.
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 927 Then, we introduce U k U k [³ 1, 1 ] and V k V k [³ 1, 1 ] for k 1, 2, 3, inductively (3.1) (3.2) U 1 [³ 1, 1 ](t) Ï e t½ ³ 1, U k [³ 1, 1 ](t) Ï for any k 2, 3,, V 1 [³ 1, 1 ](t) Ï e t½ 1, V k [³ 1, 1 ](t) Ï t k 1 k, k 1, 1 for any k 2, 3,. t k 1 k, k 1, 1 e (t )½ Ö (U k1 Ö( ½) 1 (V k2 U k2 )) d e (t )½ Ö (V k1 Ö( ½) 1 (V k2 U k2 )) d Therefore, we obtain a formal expansion u(t) U[u, Ú ](t) Ï k U k [³ 1, 1 ], k1 Ú(t) V [u, Ú ](t) Ï k V k [³ 1, 1 ], k1 as a solution to (1.1) with the initial data u ³ 1, Ú 1. Once we obtain the above expansion of the solution, the formal expansion can be justified by the use of the linear estimate of the propagator e t½ and the bilinear estimates in Section 2. Namely, we show the following result. Proposition 3.1. For any u, Ú ¾ M n,1 (Ê n ) with Ö 1 u, Ö 1 Ú ¾ M n,1 (Ê n ), there exist a small T and a unique local solution u u(t, x), Ú Ú(t, x) in C([, T ), M n,1 (Ê n )) to the Cauchy problem (1.1) with Ö 1 u, Ö 1 Ú ¾ C([, T ), M n,1 (Ê n )). Moreover they satisfy the following expansions in C([, T ), M n,1 (Ê n )): For 1, u(t) ½ k1 k U k [u, Ú ](t), Ú(t) ½ k1 k V k [u, Ú ](t), where U k [u, Ú ] and V k [u, Ú ] are defined by (3.1) and (3.2). Proof. Let U k Ï U k [u, Ú ] and V k Ï V [u, Ú ] for simplicity. We consider the case 1 since the other case is the corollary of this case. Let M, ÉM be constants satisfying (3.3) u Mn,1 Ú Mn,1 M, Ö 1 u Mn,1 Ö 1 Ú Mn,1 ÉM.
928 T. IWABUCHI AND T. OGAWA We claim that there exists C such that for k 2 and «¾ {, 1} (3.4) Ö «U k (t) Mn,1 Ö «V k (t) Mn,1 Ck 1 (k 1) (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM). 2 In the case k 2, we have for «and 1 from the smoothing effect of e t½ and (2.2) (3.5) t Ö «U 2 Mn,1 C Ö«1 e (t )½ U 1 Ö( ½) 1 (V 1 U 1 ) d Mn,1 C C C t t t (t ) (1«)2 U 1 Ö( ½) 1 (V 1 U 1 ) Mn,1 d (t ) (1«)2 U 1 Mn,1 (Ö 1 V 1 Mn,1 Ö 1 U 1 Mn,1 ) d (t ) (1«)2 d M ÉM. Ct (1 «)2 M ÉM. The estimate of V 2 is obtained in the same way as that of U 2 and we have (3.4) in the case k 2. For the constant C 1 1 satisfying the last inequality, let C of (3.4) satisfy C 2 7 C 1, and we show (3.4) in the case k 3 by induction. Let k 3 and we assume (3.4) for 2, 3,, k 1. Then, we have on the estimate of U k from the boundedness of the Riesz transform in M n,1 (Ê n ), the smoothing effect of e t½ and (2.2) (3.6) Ö «U k (t) Mn,1 C 1 t k 1 k (t ) (1«)2 U k1 Mn,1 (Ö 1 V k2 Mn,1 Ö 1 U k2 ) Mn,1 ) d. In the case 2 k j k 1 ( j 1, 2), we have from the assumption of the induction t C 1 (t ) (1«)2 U k1 Mn,1 (Ö 1 V k2 Mn,1 Ö 1 U k2 ) Mn,1 ) d C 1 t (t ) (1«)2 C k 1 1 (k 1 1) 2 ( (k 1 1)2 M ÉM k 1 1 k 1 32 M k 1 1 ÉM) C 1 ( 1) 2 ( ( 1)212 M ÉM 1 3212 M 1 ÉM) d C 1 C (k 1 1) 2 ( 1) 2t 12 «2 (t k2 12 M 2 ÉM t k22 32 M ÉM k 1 2 t k2k 12 32 M k 1 ÉM t k 52 M ÉM 2 ) 2C 1 C (k 1 1) 2 ( 1) 2 (t k2 «2 M 2 ÉM t k22 1 «2 M ÉM k 1 t k2k 12 1 «2 M k 1 ÉM t «2 M ÉM 2 ).
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 929 We apply Young s inequalities to see that all 4 terms with t, M and inequality is bounded by ÉM in the last t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM which is in the right hand side of (3.4). Therefore, we obtain (3.7) t C 1 (t ) (1«)2 U k1 Mn,1 (Ö 1 V k2 Mn,1 Ö 1 U k2 ) Mn,1 ) d 8C 1 C (k 1 1) 2 ( 1) 2 (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM). In the case (k 1, ) (1, k 1), we can not use (3.4) for k 1 1 since the power of t k 1 32 is negative. Then, we have from (3.3) for k 1 1, (3.4) for k 1 and Young s inequality (3.8) t C 1 (t ) (1«)2 U 1 Mn,1 (Ö 1 V k 1 Mn,1 Ö 1 U k 1 ) Mn,1 ) d t C 1 (t ) (1«)2 C M (k 1 1) 2 ( ()212 M ÉM C 1C 2t 12 «2 (t k2 12 M 2 ÉM t M k 1 ÉM) 2C 1C (t k2 «2 M 2 ÉM t k 32 «2 M k 1 ÉM) k 1 3212 M ÉM) d 4C 1C (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM) 16C 1 C (k 1 1) 2 ( 1) (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM). 2 Similarly, for the case (k 1, ) (k 1, 1), we also have from (3.4) for k 1 k 1,
93 T. IWABUCHI AND T. OGAWA (3.3) for 1 and Young s inequality (3.9) t C 1 (t ) (1«)2 U k 1 Mn,1 (Ö 1 V 1 Mn,1 Ö 1 U 1 ) Mn,1 ) d C 1 t (t ) (1«)2 C (k 1 1) 2 ( ()2 M ÉM k 1 32 M ÉM) ÉM d C 1C 2t 12 «2 (t k2 1 M ÉM k 1 t k 52 M ÉM 2 ) 2C 1C (t k2 12 «2 M ÉM k 1 t «2 M ÉM 2 ) 4C 1C (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM) 16C 1 C (k 1 1) 2 ( 1) (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM). 2 We take the sum of the above 3 inequalities over k 1, with k 1 k and k 1, 1. It follows from the symmetry of k 1 and that k 1 k, k 1, 1 1 (k 1 1) 2 ( 1) 2 2 2 1k 1 k2 1k 1 k2 1 (k 1 1) 2 (k k 1 1) 2 1 k 2 2 (k 2) 2 1k 1 k2 8 k2 (k 2) 2 8 (k 2) 2. 1 1 k 1 1 1 k k 1 1 2 2 k 1 1 1 x 2 dx By (3.7), (3.8), (3.9), the last inequality and C 2 7 C 1, we obtain 2 (3.1) Ö «U k (t) Mn,1 k 1 k 16C 1 C (k 1 1) 2 ( 1) 2 (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM) 8 (k 2) 2 16C 1C (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM) Ck 1 (k 1) 2 (t (k 1)2 «2 M ÉM k 1 t k 32 «2 M k 1 ÉM).
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 931 in the same way to ob- It is possible to show the same estimate as (3.1) for V k tain (3.4). Let T satisfy (3.11) C T 12 ÉM C T M 1 and T 1, È È and we have on the series k1 U k and k1 V k sup t¾(,t ) k1 M k2 M k2 ½. Ö «U k Mn,1 sup t¾(,t ) k1 Ö «V k Mn,1 C k 1 (k 1) 2 (T (k 1)2 «2 M ÉM k 1 T k 32 «2 M k 1 ÉM) M ÉM (k 1) 2 È È Then, we conclude that U Ï k1 U k and V Ï k1 V k are well defined and note that ½ U k1 U 1 U 1 U 1 U 1 U k ½ t k2 k 1 k k 1 t ½ k2 k 1 1 ½ ½ k 1 1 kk 1 1 t t e (t )½ Ö (U k1 Ö( ½) 1 (V k2 U k2 )) d e (t )½ Ö (U k1 Ö( ½) 1 (V k k1 U k k1 )) d e (t )½ Ö e (t )½ Ö (U k1 Ö( ½) 1 (V k k1 U k k1 )) d ½ k11 U k1 Ö( ½) 1 ½ 1(V k2 U k2 ) µ d U 1 t e (t )½ Ö (UÖ( ½) 1 (V U)) d, and V also satisfies the integral equation: V V 1 t e (t )½ Ö (V Ö( ½) 1 (V U)) d. This shows that (U, V ) is subject to the problem (1.1) with the initial data (u, Ú ) for small time interval [, T ).
932 T. IWABUCHI AND T. OGAWA We finally show the uniqueness of the solution in the class C([, T )Á M n,1 ). Let (u, Ú) and (Éu, ÉÚ) be solutions in C([, T )Á M n,1 ) L ½ ([, T )Á ÈM n,1 1 ) with the same initial data (u, Ú ), where ÈM n,1 1 { f ¾ S¼ Á Ö 1 f ¾ M n,1 (Ê n )}. We define R and Û(t) by R Ï Û(t) Ï By the equality sup (Ö «u() Mn,1 Ö «Ú() Mn,1 «¾{, 1},¾[,T ] Ö «Éu() Mn,1 Ö «ÉÚ() Mn,1 ), sup (Ö «(u() Éu()) Mn,1 Ö «(Ú() ÉÚ()) Mn,1 ). «¾{, 1},¾[,t] u(t) Éu(t) t and the analogous estimate to (3.5), we have e (t )½ Ö ((u Éu)Ö( ½) 1 (u Ú) ÉuÖ( ½) 1 (u Ú (Éu ÉÚ))) d Ö «(u(t) Éu(t)) Mn,1 C t (t ) (1«)2 {u Éu Mn,1 Ö 1 (u Ú) Mn,1 Ct (1 «)2 RÛ(t), Ö «(Ú(t) ÉÚ(t)) Mn,1 Ct (1 «)2 RÛ(t). It follows from the above two estimates that Éu Mn,1 (Ö 1 (u Éu) Mn,1 Ö 1 (Ú ÉÚ) Mn,1 )} d Û(t) C(t 12 t)rû(t) for all t ¾ [, T ]. Combining Û(), we obtain Û(t) if t ¾ [, T ] satisfies C(t 12 t)r 12. Repeating this procedure, we obtain the uniqueness of the solution. 4. Proof of Theorem 1.1 Let ³ ¾ S(Ê n ) be radial and satisfy and supp ³ { ¾ Ê n n} ³() 1 if 1, and let {³ j } be defined by ³ j () Ï ³( 2 j (1,,, )) for ¾ Ê n.
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 933 Let initial data {u, N } ½ N1, {Ú, N} ½ N1 be defined by u, N Ï N 12 log N Ú, N Ï N 12 log N N j(1æ)n N j(1æ)n 2 (32) j F 1 [³ j ], 2 (32) j F 1 [³ j ], where Æ satisfy Æ 17. On the estimate of U 1 [u, N, Ú, N ] and V 1 [u, N, Ú, N ], we have for p 2n and 1 ½ (4.1) U 1 [u, N, Ú, N ] ÈB 2np p, C N 12 log N C N 12 log N as N ½, V 1 [u, N, Ú, N ] ÈB 2np p, N j(1æ)n (2 ( 2np) j 2 (32) j ) 1 and we also have for p 2n and 2 ½ (4.2) U 1 [u, N, Ú, N ] ÈB 2np p, C N 12 log N V 1 [u, N, Ú, N ] ÈB 2np p, N j(1æ)n CÆ 1 N 121 log N as N ½. (2 (32) j 2 (32) j ) 1 On the estimate of {U k [u, N, Ú, N ]} k2 and {V k [u, N, Ú, N ]} k2, we use the following propositions. Proposition 4.1. Let «¾ { 1, }. Then, there exists C such that (4.3) (4.4) Ö «U 1 [u, N, Ú, N ](t) Mn,1 Ö «V 1 [u, N, Ú, N ](t) Mn,1 C2 (32«)(1Æ)N N 12 log N, Ö «U k [u, N, Ú, N ](t) Mn,1 Ö «V k [u, N, Ú, N ](t) Mn,1 C k (t (k 1)2 «2 2 (k21)(1æ)n t k 32 «2 2 ((32)k 1)(1Æ)N )N k2 (log N) k for k 2.
934 T. IWABUCHI AND T. OGAWA Proof. To show (4.3), we apply the boundedness of e t½ in M n,1 (Ê n ) to obtain Ö «U 1 [u, N, Ú, N ] Mn,1 Ö «U 1 [u, N, Ú, N ] Mn,1 C N 12 log N N j(1æ)n C2 (32«)(1Æ)N N 12 log N. 2 (32«) j (F 1 [³ j ] L n F 1 [³ j ] L n) We prove (4.4) with (3.4). Let C 1 be a constant which satisfies the above last inequality, and we take M, ÉM of (3.4) as M Ï 2C 1 2 (32)(1Æ)N N 12 log N, ÉM Ï 2C 1 2 (12)(1Æ)N N 12 log N. Then, we have (3.3) for u, N and Ú, N instead of u and Ú, and apply (3.4) to obtain (4.4). Proposition 4.2. Let ¾ S(Ê n ) Ò {} satisfy (4.5) Ç, supp Ç { ¾ Ê n 34 1 1, j 1n for j 2, 3,, n}, and let p, satisfy (1.4). Then, there exist c, C such that for t 2 2N (4.6) (4.7) U 2 [u, N, Ú, N ](t) ÈB 2np p,½ c(log N) 2 C N 1 (log N) 2, V 2 [u, N, Ú, N ](t) ÈB 2np p,½ c(log N) 2 C N 1 (log N) 2. Proof. For simplicity, let U k Ï U k [u, N, Ú, N ] and V k Ï V k [u, N, Ú, N ]. We prove (4.6) only since (4.7) is shown analogously. We have from the triangle inequality (4.8) U 2 ÈB 2np p, t t Ï I II. e (t )½ Ö (U 1 Ö( ½) 1 V 1 ) d ÈB 2np p, e (t )½ Ö (U 1 Ö( ½) 1 U 1 ) d ÈB 2np p, On the estimate of I, we have for t 2 2N and with 34 1 1 and m 1n
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 935 (m 2, 3,, n) (4.9) t e (t )½ Ö (U 1 Ö( ½) 1 V 1 ) d t e (t )2 e t2 cn 1 (log N) 2 cn 1 (log N) 2 cæ(log N) 2. Ê n e Ên 2t( ) 1 e 2( ) N j(1æ)n N j(1æ)n Then, we have from (4.9) 2 Çu, N ( ) e 2 2 ÇÚ, N () d d 2 Çu, N ( )ÇÚ, N () d d Ên 2t( ) 1 e 2( ) 2 2(32) j ³ j ( )2(32) j ³ j () d 2 3 j 2 3 j ³ j ( )³ j () d Ê n (4.1) I cæ(log N) 2. On the estimate of II, we have from the embedding È B 1 n,1 (Ên ) È B 2np p, (Ê n ), (2.3) and (2.4) (4.11) II C C C C C t t t t t e (t )½ Ö U 1 Ö( ½)U 1 ÈB 1 n,1 d e (t )½ U 1 Ö( ½)U 1 ÈB n,1 d (t ) 12 U 1 Ö( ½)U 1 ÈB 1 n,½ d (t ) 12 U 1 2 B È 1 d 2n,1 (t ) 12 12 u, N 2 B È 32 d 2n,½ C N 1 (log N) 2. Therefore, we obtain (4.6) by (4.8), (4.1) and (4.11). We consider the sequence {(u N, Ú N )} ½ N1 of the solutions which are expanded by u N (t) ½ U k [u, N, Ú, N ](t), Ú N (t) ½ k1 k1 V k [u, N, Ú, N ](t),
936 T. IWABUCHI AND T. OGAWA with the initial data u N () u, N and Ú N () Ú, N. Let ¾ S(Ê n ) be a function satisfying (4.5). We consider the solution at time t 2 2N and have from triangle inequality, (4.6) and U 1, u N (t) ÈB 2np p, u N (2 2N ) ÈB 2np p, (4.12) c u N (2 2N ) ÈB 2np p, c U 2 [u, N, Ú, N ](2 2N ) ÈB 2np c U p, 1 [u, N, Ú, N ](2 2N ) ÈB 2np p, c k3 U k [u, N, Ú, N ](2 2N ) ÈB 2np p, c(log N) 2 C N 1 (log N) 2 C U k [u, N, Ú, N ](2 k3 2N ) ÈB 2np. p, Then, we have from supp Ç being compact and away from the origin, (4.4) and t 2 2N k3 C U k [u, N, Ú, N ](t) ÈB 2np p, C k3ö 1 U k [u, N, Ú, N ](t) Mn,1 (4.13) k3 C k (t (k 1)212 2 (k21)(1æ)n t k 3212 2 ((32)k 1)(1Æ)N )N k2 (log N) k k3 C k (2 k N 2 (k21)(1æ)n 2 2(k 1)N 2 ((32)k 1)(1Æ)N )N k2 (log N) k. Since Æ 17, we have 2 k N 2 (k21)(1æ)n 2 2(k 1)N 2 ((32)k 1)(1Æ)N 2 if k 3. Then, it follows from (4.13) that k3 C U k [u, N, Ú, N ](2 2N ) ÈB 2np p, as N ½. Therefore, we have from (4.12) and the last estimate on the solution at time t 2 2N u(2 2N ) ÈB 2np p, ½ as N ½. The divergence of Ú at time t 2 2N the proof of Theorem 1.1. is obtained in the same way and we complete
ILL-POSEDNESS FOR THE DRIFT DIFFUSION SYSTEM 937 5. Concluding remark on Navier Stokes system We should mention finally that similar structures to (1.5) exist for the incompressible Navier Stokes equations and the vorticity equations. For the incompressible Navier Stokes equations t u ½u (u Ö)u Ö p, t, x ¾ Ê 3, div u, t, x ¾ Ê 3, u(, x) u (x), x ¾ Ê 3, the scaling invariant Besov spaces are B È 13p p,q (Ê 3 ) with 1 p,q ½. One can regard the case p 3 ( n) for the Navier Stokes equations as the case p 32 ( n2) for the drift diffusion system (1.1) in the Besov spaces B È 2np p,q (Ê n ) since the scaling invariant Lebesgue space for the incompressible Navier Stokes equations and the drift diffusion equations is L 3 (Ê 3 ) and L 32 Ê 3 ), respectively. The well-posedness for the Navier Stokes equations in B È 13p p,½ (Ê 3 ) (3 p ½) was considered in Kozono Yamazaki [16], Cannone Planchon [4], and the ill-posedness in B È 1 ½,½ (Ê3 ) was shown by Bourgain Pavlović [2] and Yoneda [29]. Therefore, the case p ½ is optimal for the well-posedness and the ill-posedness on the study of the Navier Stokes equations, and the important structure of nonlinear term is divu and (u Ö)u Ö (u Å u), which corresponds to the structure (1.5) of the equation (1.2). For the vorticity equations: t ½ (u Ö) ( Ö)u, t, x ¾ Ê 3, div, t, x ¾ Ê 3, (, x) rot u (x), x ¾ Ê 3, scaling invariant Besov spaces for the vorticity are B È 23p p, (Ê 3 ) (1 p, ½) since rot u. Critical case should be p ½ since the case p ½ for the Navier Stokes equation is optimal for the well-posedness and the ill-posedness. In those system, the nonlinear structure is again has a special symmetry and one can find that the nonlinearity can be expressed by (u Ö) ( Ö)u Ö 2 {(Ö 2 )} similar to (1.5). Indeed, the first component of the nonlinear term can be seen that 3 j1 (u j x j 1 j x j u 1 ) 3 j1 3 j1 x j (u j 1 j u 1 ) j [{( ½) 1 ( xk l xl k )} 1 j ( ½) 1 ( x2 3 x3 2 )],
938 T. IWABUCHI AND T. OGAWA where we used div u div, the Bio Savart law u ( ½) 1 rot, and ( j, k, l) ¾ {1, 2, 3} 3 satisfy the property of cyclic change, namely j 2 k 1 l (mod 3). For simplicity, let I j Ï {( ½) 1 ( xk xl xl k )} 1 j ( ½) 1 ( x2 3 x3 2 ). If j 1, we have I 1 {( ½) 1 ( x2 3 x3 2 )} 1 1 ( ½) 1 ( x2 3 x3 2 ). If j 2, we have from div I 2 {( ½) 1 ( x3 1 x1 3 )} 1 2 ( ½) 1 ( x2 3 x3 2 ) 1 ( ½) 1 x3 1 2 ( ½) 1 x3 2 3 ( ½) 1 x3 3 ( ( ½) 1 Ö) 3 1 ( ½) 1 x3 1 2 ( ½) 1 x3 2 3 ( ½) 1 x3 3 Ö (( ½) 1 3 ). The first, second and third terms of the last right hand side can be regarded as Ö(Ö 2 ) analogous way of (1.5) and we can regard x2 I 2 as Ö 2 (Ö 2 ). The case j 3 is also treated in the similar way to the case j 2. The other components are also treated analogously, and therefore we obtain the structure (u Ö) ( Ö)u Ö 2 {(Ö 2 )} similarly to (1.5). References [1] P. Biler and J. Dolbeault: Long time behavior of solutions of Nernst Planck and Debye Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2), 461 472. [2] J. Bourgain and N. Pavlović: Ill-posedness of the Navier Stokes equations in a critical space in 3D, J. Funct. Anal. 255 (28), 2233 2247. [3] P. Biler, M. Cannone, I.A. Guerra and G. Karch: Global regular and singular solutions for a model of gravitating particles, Math. Ann. 33 (24), 693 78. [4] M. Cannone and F. Planchon: Self-similar solutions for Navier Stokes equations in R 3, Comm. Partial Differential Equations 21 (1996), 179 193. [5] L. Corrias, B. Perthame and H. Zaag: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (24), 1 28. [6] W. Fang and K. Ito: Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations 123 (1995), 523 566. [7] H.G. Feichtinger: Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983, in; Proc. Internat. Conf. on Wavelets and Applications, New Delhi Allied Publishers, 1 56, 23. [8] H. Gajewski and K. Gröger: On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), 12 35. [9] T. Iwabuchi: Global well-posedness for Keller Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (211), 93 948. [1] T. Iwabuchi and T. Ogawa: Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc. 367 (215), 2613 263. [11] A. Jüngel: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci. 5 (1995), 497 518. [12] E.F. Keller and L.A. Segel: Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (197), 399 415.
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