Remarks on the ill-posedness results for the drift diffusion system (Harmonic Analysis and Nonlinear Partial Differential Equations)

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(1)Title. Author(s). Citation. Issue Date. Remarks on the ill-posedness results for the drift diffusion system (Harmonic Analysis and Nonlinear Partial Differential Equations) Iwabuchi, Tsukasa; Ogawa, Takayoshi. 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu (2016), B56: 31-41. 2016-04. URL. http://hdl.handle.net/2433/241320. Right. © 2016 by the Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.. Type. Departmental Bulletin Paper. Textversion. publisher. Kyoto University.

(2) RIMS Kôkyûroku Bessatsu B56 (2016), 31−41. Remarks. the. on. ill‐posedness. results. for the drift diffusion system By. Tsukasa IWABUCHI. *. and. Takayoshi OGAWA. **. Abstract. The. ill‐posedness. issue is considered for the drift‐diffusion. shown that the continuous. dependence. on. system of bipolar type. It is. initial data does not hold. generally. in the. scaling. \dot{B}_{p, $\sigma$}^{-2+\frac{n}{p} (\mathb {R}^{n}). invariant Besov spaces. The. with 1 \leq p, $\sigma$\leq 1, scaling invariant Besov spaces are of the case is to obtain the well‐posedness p=2n explained optimality and the ill‐posedness for the drift diffusion system of bipolar type with the attention to the divergence form structure of the nonlinear term. The scaling invariant spaces for the heat equation with the nonlinear term u^{2} are same as those for the drift diffusion system and the and the. reason on. the. difference is also indicated. on. the. ill‐posedness. equation and the drift. results between the heat. diffusion system.. Introduction. §1. The purpose of this paper is. drift diffusion equation of results for other. surveying. bipolar type. equations. with. the result in. [11]. on. the. ill‐posedness for. in the Besov spaces to compare the. quadratic. nonlinear terms such. as. scaling. the. ill‐posedness. invariant spaces. are. common.. results among these. We shall. equations due. explain. to the. ill‐posedness. the drift diffusion. equation of monopolar type, and the heat equation with the nonlinear the. the. term. u^{2} where ,. the essential difference of. divergence. form structure of. nonlinear terms. Received. September 30, 2014. Revised April 21, 2015. Subject Classification(s): 35\mathrm{K}08, 35\mathrm{K}55 Key Words: Ill‐posedness, Besov spaces, Drift diffusion equation. Osaka City University, Osaka 558‐8585, Japan.. 2010 Mathematics *. \mathrm{e} ‐mail: **. iwabuchi@sci. osaka‐cu.. jp 980‐8578, Japan. \mathrm{e} ‐mail: ogawa@math. tohoku. ac. jp Tohoku. ©. University,. ac.. Sendai. 2016 Research Institute for Mathematical. Sciences, Kyoto University.. All. rights. reserved..

(3) Tsukasa Iwabuchi. 32. Let. us. and. Takayoshi Ogawa. problems of. consider the initial value. drift‐diffusion equation of. a. bipolar. type:. \left{bginary}{l \partil_{}u-\rangleu+$\kap nbla\cdot(unabl$\psi)=0,&t>x\inmathb{R}^n,\ partil_{}v-\anglev-$\kap nbla\cdot(vnabl$\psi)=0,&t>x\inmathb{R}^n,\ -triangle$\psi=v-u,&t>0x\inmathb{R}^n,\ u(0x)=_{}(,v0x)=_{}(,&x\inmathb{R}^n, \ed{ary}\ight.. (1.1). where. and. u. constant and. coupling. (1.1). the. v are. was. Neumann. originally. particle density of negative. we assume. $\kappa$=\pm 1. u_{0} and v_{0}. considered for condition. boundary. semi‐conductor device and. we. as. This model is also the. an. initial. the. refer to. model of the semiconductor device. (1.2). simulation,. regarded. the. as. 23, 29] and references theirin. limitting. This system. on. problems. 2n). p <. .. [31]. showed. Iwabuchi. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). case. was. [9]. showed. obtained the .. See also. that the continuous. system. for. equations. of the. scaling. in. problem. system. known. as. well‐posedness. the similar line. For. issue. instance,. global. on. was. the. solutions. spaces and. in the. weighted Lebesgue Hardy space \mathcal{H}^{1} (R2).. in the Besov spaces. for the. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). larger. (n/2<. Besov spaces. [15], [16].. dependence. on. ,. ill‐posedness for. the. problem (1.1) by. initial data does not hold. generally. where there is the invariant. scaling. in the to the. $\lambda$>0,. (1.3). the. we. global well‐posedness. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}) (2n \leq p \leq \infty). (1.1), namely. considered;. by Keller‐Segel [13] in 1970, refer to [3, 5, 9, 15, 16, 21, 22,. global well‐posedness. global well‐posedness. n/2\leq p<1. Besov spaces. model is also. of the chemotaxis. Lebesgue. in. a. for the related results. As the. monopolar. and. along. are. The purpose of this paper is to prove the. showing. models for simulation of. or. introduced. molds,. by Kurokiba‐Ogawa [18, 19]. Zhao‐Liu‐Ciu. the. is the. with Dirichlet. in whole space, the existence of local solutions and. Ogawa‐Shimizu [23]. spaces.. simplest. $\kappa$. initial data. The system. problem. the results of the system. The. considered in both models and results. shown. given. value. [1, 6, 8, 12, 20, 25]. describes the motion of the chemotaxis. was. are. boundary. of the. one. positive electric charge,. \left{\begin{ar y}{l \partil_{}u-\triangleu+$\kap $\nabl\cdot(u\nabl $\psi$)=0,t> x\in mathb{R}^n,\ -triangle$\psi$=u,t>0x\in mathb{R}^n,\ u(0,x)=u_{0}(x), \in mathb{R}^n \end{ar y}\right.. Keller‐Segel system [13].. initial value. and. (1.1). \left{bginary}{l u_$\ambd}(t,x)=$\lambd^{2}u($\lambd^{2}t,$\lambdx),\ v_{$lambd}(t,x)=$\lambd^{2}v($\lambd^{2}t,$\lambdx),\ $psi_{\lambd$}(t,x)=\psi$(lambd$^{2}t,\lambd$x), \end{ary}\ight.. is maintained invariant.. transformations between u_{ $\lambda$}, v_{ $\lambda$} and. $\psi$_{ $\lambda$}. Here,. we. comes. note that the difference. from the third equation.

(4) Remarks. -\triangle $\psi$. =v-u. L^{ $\theta$}(\mathbb{R}_{+};X) is the. (1.1).. in. with. Lebesgue. 33. on the 1LL‐Posedness results for the drift diffusion system. The invariant function space is defined. scaling (1.3). Banach scale X , where the invariant. a. space. L^{p}(\mathbb{R}^{n}). ( $\theta$, p) necessarily. then the exponents. ,. the Bochner space. by. is invariant. If X. satisfies. 2=\displaystyle\frac{2}{$\theta$}+\frac{n}{p} so. L^{ $\theta$}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{n}). that. property holds and. we. is the invariant space.. a. critical space. can. be. indeces. negative regularity properties. the. Indeed,. rature.. as. far. \dot{B}_{p, $\sigma$}^{-2+\frac{n}{p} (Rn).. that there appears. a. enough. is not. Riesz. scaling. monopolar type the. ill‐posedness. p=1 is threshold for the. the. in the. specify. the critical space for the. 1. introduce the. was. displaystyle \frac{n}{p}(\mathb {R}^{n}) B_{p,-2\+\infty}. shown in Iwabuchi. monopolar type (1.2). in the. and. [9].. well‐posedness,. space. scaling. \dot{H}_{p}^{s}. (. with. invariant Besov. \displaystle\frac{n}2. < p <. of the. the and. 1. This stands that the. sense. it. with the. problem (1.2) of. The initial value. in the Besov spaces. bipolor. issue in the function spaces. homogeneous Sobolev. below.. ,. with. shall show. we. ,. initial data between the. To. we. well‐posedness. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). p\geq 2n. case. well‐posedness. \dot{B}1_{\infty}^{2}(\mathbb{R}^{n}). \in. initial data in the known litu‐. for the. with 1 \leq p \leq. in. (u, v, $\psi$). as. and. are common. regular. regularity. and 1 \leq p\leq\infty ) and. well‐posed. is. (1.2). and. (1.2). However,. and. to describe in term of the. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). consistency. Besov spaces with. homogeneous. (1.2). is invariant.. potential of s\in \mathbb{R}. spaces. and. the. (1.1). is obtained in the Besov spaces. difference for low. monopolar systems (1.1) where the. (1.1). (1.1). consider rather. well‐posedness. p< 2n for both equations. term of the. generalized by. as we. ,. with the boundedness of the solution in time variable.. Almost all basic nature of the models coinsides. 1. =. reach the invariant function spaces for. L^{n/2}(\mathbb{R}^{n})\times L^{n/2}(\mathbb{R}^{n})\times BMO(\mathbb{R}^{n}) Such. particular $\theta$. In. case. well‐posedness. and. ill‐posedness. On the other hand for the equation .. that the. problem. is. well‐posed. the critical space for the is identified. as. p. =. 2n. through .. of. B_{p,\infty} p(\mathbb{R}^{n}) the. bipolar type, Zhang‐Liu‐Ciu [31] with. and the. well‐posedness. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}) (2n \leq p \leq \infty) 2n\leq p_{2}. in. (1.1). -2+\mathrm{n}. study. We note the. \displaytle\frac{n}2. <p< 2n. .. ill‐posedness. following. inclusion. we. show that. equation (1.1). to the. ill‐posedness. of the. Then,. showed. in the Besov spaces. relations; for \displaytle\frac{n}2 \leq. p_{1}. \leq. it holds that. L^{\frac{n}{2} (\mathb {R}^{n})\subset\dot{B}_{p_{1},\infty}^{-2+\frac{n}{p_{1} (\mathb {R}^{n}) \subset\dot{B}_{2n,\infty}^{-\frac{3}{2} (\mathb {R}^{n}) \subset\dot{B}_{p_{2},\infty}^{-2+\frac{n}{p_{2} (\mathb {R}^{n}) \subset\dot{B}_{\infty,\infty}^{-2}(\mathb {R}^{n}) We define the. homogeneous Sobolev. We denote the function spaces of distributions. by S'(\mathbb{R}^{n}). ,. and. Definition (the homogeneous. and the Besov spaces and state. rapidly decreasing. polynomials by \mathcal{P}(\mathbb{R}^{n}) Sobolev. spaces).. functions. our. by S(\mathbb{R}^{n}). ,. theorems.. tempered. .. For any. s\in \mathbb{R},. 1. \leq p\leq. 1,. the. homoge‐.

(5) Tsukasa Iwabuchi. 34. neous. Sobolev space. \dot{H}_{p}^{s}=\dot{H}_{p}^{s}(\mathbb{R}^{n}) where. \dot{H}_{p}^{s}(\mathb {R}^{n}). :=. \{f\in S(\mathb {R}^{n})/\mathcal{P}(\mathb {R}^{n})|\Vert f\Vert_{\dot{H}_{p}^{\mathrm{s} } := \Vert \mathcal{F}^{-1}[| $\xi$|^{s}\hat{f}( $\xi$)]\Vert_{L^{p}(\mathb {R}^{n})} <1\},. \mathcal{F}^{-1} denotes the. inverse Fourier. transform.. spaces).. Besov. Let. transform of $\phi$. is the Fourier. and let. ,. $\phi$_{j}(x) :=2^{nj} $\phi$(2^{j}x) Then, for. $\phi$\in S(\mathbb{R}^{n}). \hat{ $\phi$}\subset\{ $\xi$\in \mathb {R}^{n}|2^{-1} \leq | $\xi$| \leq 2\}, \displaystle\sum_{j\in mathb{Z}\hat{$\phi$}(\frac{$\xi}{2^j}). supp. \hat{$\phi$}. Takayoshi Ogawa. defined by. is. Definition (the homogeneous. where. and. any. s. \in \mathbb{R},. 1. \leq p,. \leq. $\sigma$. the. 1,. \{$\phi$_{j}\}_{j\in \mathbb{Z}. be. =1. satisfying. the. following:. for $\xi$\in \mathbb{R}^{n}\backslash \{0\},. defined by. be. for j\in \mathbb{Z}, x\in \mathbb{R}^{n}. homogeneous. Besov space. \dot{B}_{p, $\sigma$}^{s}(\mathb {R}^{n}). is. defined. by. \dot{B}_{p, $\sigma$}^{s}(\mathb {R}^{n}) Remark.. of. S'(\mathbb{R}^{n}). :=. \{f\in S(\mathb {R}^{n})/\mathcal{P}(\mathb {R}^{n})|\Vert f\Vert .p, $\sigma$ \mathrm{s} := \Vert\{2^{sj}\Vert$\phi$_{j}*f\Vert_{L^{p}(\mathb {R}^{n})}\}_{j\in \mathb {Z} \Vert_{\el ^{ $\sigma$}(\mathb {Z})} <1\}.. It is. for. Sobolev space. possible. some. s, p, q. \dot{H}_{p}^{s}(\mathb {R}^{n}). is. regard. to. the above. Indeed, if. .. equivalent. s. <. n/p. equivalent. with 1 \leq q \leq. 1. ,. or s. =. <1,. n/p. equivalence. are. Theorem 1.1.. (1.4). satisfy. s. n/p. <. ,. certain. then the. subspaces. homogeneous. u=\displaystyle \sum_{j\in \mathb {Z} $\phi$_{j}*u. with q. =. in. S'(\mathbb{R}^{n}) }.. 1 , the Besov space. \dot{B}_{p,q}^{s}(\mathbb{R}^{n}). is also. to. { u\in S'(\mathbb{R}^{n})|\Vert u\Vert_{\dot{B}_{p,q}^{\mathrm{s} } These. and p. as. spaces. to. { u\in S'(\mathbb{R}^{n})|\Vert u\Vert_{\dot{H}_{p}^{\mathrm{s} } If. s. homogeneous. <1,. due to the argument. [11]. Let. n\geq 2,. u=\displaystyle \sum_{j\in \mathb {Z} $\phi$_{j}*u by. Kozono‐Yamazaki. $\kappa$=\pm 1 and let p,. 2n<p\leq 1 and 1\leq $\sigma$\leq 1. ,. S'(\mathbb{R}^{n}) }.. in. or. $\sigma$. [17].. satisfy. p=2n and 2< $\sigma$\leq 1.. \{T_{N}\}_{N} with T_{N} \rightarrow 0(N\rightarrow\infty) and a sequence of smooth and rapidly decreasing initial data \{u_{0,N}\}_{N}, \{v_{0,N}\}_{N} (N = 1,2, \cdots) such that the corresponding sequence of smooth solutions \{u_{N}\}_{N}, \{v_{N}\}_{N} to (1.1) with u_{N}(0) u_{0,N} and v_{N}(0)=v_{0,N} satisfies Then,. there exist. a. sequence. of. time. =. \displaystyle \lim_{N\rightar ow\infty}\Vert u_{0,N}\Vert p,$\sigma$-2+\displaystyle\frac{n}{p} =0, \displaystyle \lim_{N\rightar ow\infty}\Vert v_{0,N}\Vert p,$\sigma$-2+\displaystyle\frac{n}{p} =0, \displaystyle \lim_{N\rightar ow\infty}\Vert u_{N}(T_{N})\Vert p,$\sigma$-2+\displaystyle\frac{n}{p} =\infty, \displaystyle \lim_{N\rightar ow\infty}\Vert v_{N}(T_{N})\Vert p,$\sigma$-2+\displaystyle\frac{n}{p} .. .. .. .. =\infty..

(6) Remarks. Solutions in the above theorem. Remark.. well‐posed. ,. (u_{N}, v_{N}). ,. by. the result. [31],. We note that when. global well‐posedness. in this paper.. locally. (u_{N}, v_{N}). and the solution. is. parameter $\epsilon$>0. a. and the. on. the. \dot{B}_{4}^{-\frac{3}{2 } (\mathb {R}^{2}). $\sigma$=2 , the Besov space. It is indeed. ill‐posedness.. for small initial data in the. [11]. introduce. we. n=2, p=4 and. well‐posedness. We refer to the paper. same. possible. to. space.. precise proof, and explain. an. idea of the. (1.1) by. formal expansion of the solution to. some. proof small. as. (1.5). u=U_{0}+ $\epsilon$ U_{1}+$\epsilon$^{2}U_{2}+$\epsilon$^{3}U_{3}+\cdots. (1.6). v=V_{0}+ $\epsilon$ V_{1}+$\epsilon$^{2}V_{2}+$\epsilon$^{3}V_{3}+\cdots we. is. where T_{N} denotes the maximal existence time of the solution. is the critical space for the. Then,. problem (1.1). smooth. The. are. and T_{N}<T_{N} for all N.. Remark.. show the. \dot{B}_{\underline{n},1}^{-1}(\mathb {R}^{n}). in the Besov space. [0,\overline{T}_{N}) \dot{B}_{n,1}^{-1} (Rn)),. in C (. 35. on the 1LL‐Posedness results for the drift diffusion system. regard. the parameter. rameter N for the initial data.. $\epsilon$. depending only. Namely. we. on. ,. :. the inverse of the. u_{0} and v_{0}. expand. u_{0}=$\varphi$_{0}+ $\epsilon \varphi$_{1}+$\epsilon$^{2}$\varphi$_{2}+$\epsilon$^{3}$\varphi$_{3}+\cdot \cdot \cdot. frequency. pa‐. by. ,. v_{0} =$\rho$_{0}+ $\epsilon \rho$_{1}+$\epsilon$^{2}$\rho$_{2}+$\epsilon$^{3}$\rho$_{3}+\cdot \cdot \cdot by $\epsilon$\cong N^{-1}. (1.6). and consider the. of the solution to. (1.1). the initial data which low. equations for U_{k}. =U_{k}[u_{0}, v_{0}],. with the initial data. frequency part. component $\varphi$_{1} and $\rho$_{1} the initial data ,. is. U_{k}(0). vanishing. u_{0} and v_{0}. are. V_{k} $\varphi$_{k},. =. =V_{k}[u_{0}, v_{0}] V_{k}(0). and it has. .. ,. only if. u. \cdots. ,. $\epsilon$ :. $\epsilon$^{2}. :. :. ,. \left{begin{ar y}l (\partil_{}-\triangle)U_{1}=0,\ (partil_{}-\triangle)V_{1}=0,\ U_{1}(0)=$\varphi$_{1},V (0)=$\rho_{1}, \end{ar y}\ight. \left{\begin{ar y}{l (\partil_{}-\triangle)U_{2}=-$\kap $\nabl.(U_{1}\nabl(-\triangle)^{-1}(V_{ -U_{1}),\ (partil_{}-\triangle)V_{2}=$\kap $\nabl.(V_{1}\nabl(-\triangle)^{-1}(V_{1}-U_{1}),\ U_{2}(0)=,V_{2}(0)=, \end{ar y}\ight. :. =$\rho$_{k}. .. some. (1.5). If. we. and take. frequency. given by u_{0}=0+ $\epsilon \varphi$_{1}+$\epsilon$^{2}\times 0+\cdots. v_{0}=0+ $\epsilon \rho$_{1}+$\epsilon$^{2} \times 0+\cdots and we find U_{0}\equiv V_{0}\equiv 0 Then, equation (1.1), we have by the term of order $\epsilon$^{k} with k=1 2, 3, and. in. and. such. v. as. satisfy. the.

(7) Tsukasa Iwabuchi. 36. In. respectively.. $\epsilon$^{k}. (1.7). general,. 2, 3,. \cdots. have. the term of order k\geq 2. on. \left{bginary}{l (\partil_{}-\riangle)U_{k}=-$\ap$\sum_{k1}+_{2=k, 1}_{2\geq1}(U_{k1}\nabl(-triangle)^{-1}(V_k{2}-U_k{2}),\ (partil_{}-\riangle)V_{k}=$\ap$\sum_{k1}+_{2=k, 1}_{2\geq1}(V_{k1}\nabl(-triangle)^{-1}(V_k{2}-U_k{2}),\ U_{k}(0)=,V_{k}(0)=, \end{ary}\ight.. :. formally. Therefore, k=1 ,. we. Takayoshi Ogawa. and. it is natural to introduce. U_{k}. U_{k}[$\varphi$_{1}, $\rho$_{1}]. =. and V_{k}. =. V_{k}[$\varphi$_{1}, $\rho$_{1}]. for. inductively. (1.8). \left{begin{ar y}l U_{1}[$\varphi$_{1},\rho$_{1}](t):=e^{t\riangle}$\varphi$_{1},\ U_{k}[$\varphi$_{1},\rho$_{1}](t):=-$\kap $\sum_{k1}+_{2=k,_{1}k2\geq1}\int_{0}^ e{(t-$\au)triangle}\abl\cdot(U_{k1}.\nabl(-\triangle)^{-1}(V_{k2}-U_{k2})d$\tau \ mathr{f}\mathr{o}\mathr{}\mathr{a}\mthr{n}\mathr{y}k=2,3\cdots, \end{ar y}\ight. \left{begin{ar y}l V_{1}[$\varphi$_{1},\rho$_{1}](t):=e^{t\riangle}$\rho_{1},\ V_{k}[$\varphi$_{1},\rho$_{1}](t):=$\kap $\sum_{k1}+_{2=k,_{1}k2\geq1}\int_{0}^ e{(t-$\au)triangle}\abl\cdot(V_{1}.\nabl(-\triangle)^{-1}(V_{k2}-U_{k2})d$\tau \ mathr{f}\mathr{o}\mathr{}\mathr{}\mathr{n}\mathr{y}k=2,3\cdots: \end{ar y}\ight.. (1.9). In view of the. expansion (1.5) and (1.6),. we. consider. u(t) = U_{0}[0, 0](t) + \displaystyle \sum$\epsilon$^{k}U_{k}[$\varphi$_{1}, $\rho$_{1}] = \sum$\epsilon$^{k}U_{k}[$\varphi$_{1}, $\rho$_{1}], k=1. k=1. v(t) =V0 [0, 0](t) + \displaystyle \sum$\epsilon$^{k_{V_{k} }[$\varphi$_{1}, $\rho$_{1}] = \sum$\epsilon$^{k_{V_{k} }[$\varphi$_{1}, $\rho$_{1}], k=1. solution to. as a. (1.1). Under such the solution. One. simplest. for the initial data u_{0}= $\epsilon \varphi$_{1}, v_{0}= $\epsilon \rho$_{1}.. formal expansion,. stay modest way. can. ported locally supp. a. at. some. \cong. \{ $\xi$;| $\xi$| \cong 2^{N}\} higher. that each term in the. investigate. diverge. as. t \rightarrow. by. 0. some. expansion of. initial test functions.. for N. 1,. =. test function. a. 2,. \cdots .. expansion. By. ,. N. \rightarrow. \rightarrow. Then. blow up in the. larger. the. expansion parameter. some. space. $\epsilon$ as a. u_{0,N}, v_{0,N} with. those test. functions,. one. lower order term such. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). 0 , where p_{0} is the threshold index for the. fixing. as. U_{k}[u_{0,N}, v_{0,N}], V_{k}[u_{0,N}, v_{0,N}]. k_{0} remains bounded under N\rightarrow 1 while. U_{2}[u_{0,N}, v_{0,N}], V_{2}[u_{0,N}, v_{0,N}] with t. We let such. terms from the. as. ill‐posedness.. or. particular frequency.. may find that all the. 1. we. test function is the monochromatic data that Fourier transform is sup‐. \overline{u_{0,N} , \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\overline{v_{0,N}. k\geq k_{0} for. k=1. for p>p_{0}. as. well‐posedness. and the. the. equation. constant,. we see.

(8) Remarks. is. the. (1.1). of. case. \geq p_{0}. p. there appears. a. For the system. .. is p. 1. =. (1.2). On the other. .. saturation because of the difference. one can. not attain the. of. well‐posedness. on. up. p=1.. The main. (1.2). bipolar type,. or. well‐posedness. form of the nonlinear part, and. divergence. to the. with p > p_{0}. the threshold index for the. monopolar type, hand for. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (\mathb {R}^{n}). in the space. ill‐posed. 37. on the 1LL‐Posedness results for the drift diffusion system. reason. why. the. is that the nonilnear structure of each. its derivative structure. For the. u\nabla(-\triangle)^{-1}u. satisfies the. (1.1). class differs between the system. well‐posedness. system has. and. essential difference with. an. equation (1.2) of monopolar type, the nonlinear. term. following identity:. \displaystyle \partial_{x_{j} u\partial_{x_{j} (-\triangle)^{-1}u= \frac{1}{2}\partial_{x_{j} (-\triangle)\{( -\triangle)^{-1}u)(\partial_{x_{j} (-\triangle)^{-1}u)\} +\partial_{x_{j}}\nabla\cdot\{((-\triangle)^{-1}u)(\nabla\partial_{x_{j}}(-\triangle)^{-1}u)\}. (1.10). +\displaystyle \frac{1}{2}\partial_{x_{j} ^{2}\{( -\triangle)^{-1}u)u\}, which. term \nabla. (u\nabla(-\triangle)^{-1}u). shows that the. [9].. observed in. was. we. may. divergence. such. regard. as. important point. is that. bipolar type (1.1),. we can. the nonlinear. regard. |\nabla|^{2}\{(|\nabla|^{-2}u)u\}. the nonlinear term for. roughly by (1.10). This expression the monopolar type can be treated as. form of the second order. This enables. weaker Besov space up to the. The most. \dot{B}_{p,\infty}^{-2+\frac{n}{p}. with 2 \leq p <. and. 1. to treat the. us. \dot{B}_{\infty,2}^{-2}. (u\nabla(-\triangle)^{-1}v). the nonlinear term \nabla.. in the nonlinear structure which prevents to have such. a. .. in the. equation. On the other hand for. has lack of the symmetry. good expression. is. as. seen. in. (1.10). To be e_{1}. :=. more. precise, let. (1, 0, \cdots , 0). ,. and. we. $\chi$ be. a. characteristic function which support is. take initial data. We note that the Fourier transforms of u_{0} and v_{0}. frequency where. N and -N ,. \dot{H}_{p}^{s}(\mathb {R}^{n}). is the. respectively,. and. and. as. u_{0}=N^{2-\frac{n}{p} \mathcal{F}^{-1}[ $\chi$(\cdot-N \mathrm{e}1)], v_{0}=N^{2-\frac{n}{p} \mathcal{F}^{-1}[ $\chi$(\cdot+N \mathrm{e}1)]. (1.11). [−1, 1]. \Vert u_{0}\Vert. homogeneous Sobolev. .. are. p-2+\displaystyle \frac{n}{p},. space.. supported locally. \Vert v_{0}\Vert. By. .. p-2+\displaystyle \frac{n}{p}. :. at. particular. independent of N,. are. the Duhamel. formula,. we. write. the solution. u(t)=e^{t\triangle}u_{0}- $\kap a$\displaystyle \int_{0}^{t}e^{(t- $\tau$)\triangle}\nabla. (u\nabla(-\triangle)^{-1}(v-u) d $\tau$, v(t)=e^{t\triangle}v_{0}+ $\kap a$\displaystyle \int_{0}^{t}e^{(t- $\tau$)\triangle}\nabla. (v\nabla(-\triangle)^{-1}(v-u) d $\tau$. Since both of. Then,. we. u. and. v. approximate. are. treated. similarly,. the nonlinear part in. we. consider the nonlinear term of. right. hand side. by. the. use. u. only.. of the approxi‐.

(9) Tsukasa Iwabuchi. 38. mation with the linear parts. Takayoshi Ogawa. and. u\cong e^{ $\tau$\triangle}u_{0}, v\cong e^{ $\tau$\triangle}v_{0}. to. see. that. (1.12). \displaystyle\Vert\int_{0}^{t}e^{(t-$\tau$)\triangle}\nabla. (e^{ $\tau$\triangle}u_{0}\nabla(-\triangle)^{-1}e^{t\triangle}(v_{0}-u_{0}) d $\tau$\Vert_{\dot{H}_{p}^{-2+\frac{n}{p} =. \cong. \cong. \displaystyle\Vert\mathcal{F}^{-1}[|$\xi$|^{-2+\frac{n}{p} $\xi$\cdot\int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)(\hat{v}_{0}($\eta$)-\^{u}0($\eta$) \frac{$\eta$}{|$\eta$|^{2}\frac{e^{-t|$\xi$|^{2}(e^{2t($\xi$-$\eta$)\cdot$\eta$}-1)}{2($\xi$-$\eta$)\cdot$\eta$}d$\eta$]\Vert_{L^{p} \displaystyle\Vert\mathcal{F}^{-1}[|$\xi$|^{-2+\frac{n}{p} $\xi$\cdot\int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)\hat{v_{0} ($\eta$)\frac{N}{N^{2} \frac{1}{2N^{2} d$\eta$]\Vert_{L^{p} \Vert \mathcal{F}^{-1}[| $\xi$|^{-2+\frac{n}{p} $\xi \chi$( $\xi$)]\Vert_{L^{p} N^{2(2-\frac{n}{p})}N^{-1}N^{-2}. \cong N^{1-\frac{2n}{p} , diverges as N\rightarrow 1 if p>2n In the third line of (1.12), we suppose convolution of û0 and \hat{v_{0} should have less regulariy than that of û0 and itself due. and the last term that the. .. to the difference of the. û0. divergence. form structures of two terms to. the convolution. ignore. frequency restriction | $\xi$| \leq 2, | $\xi$- $\eta$| \simeq N and | $\eta$| \simeq N obtained for $\xi$- $\eta$\in supp û0, $\eta$\in supp \hat{v_{0} with the initial data given by (1.11). On the other hand for the part with the convolution of û0 and û0, we have from the structure \nabla\cdot(u\nabla(-\triangle)^{-1}u)\cong |\nabla|^{2}\{(|\nabla|^{-2}u)u\} by (1.10) and the support of û0 * û0 being in the of. and. itself,. neighborhood. and. we. of the. utilized the. frequency. 2N ,. namely, $\xi$- $\eta$\simeq N, $\eta$\simeq N. and. $\xi$\simeq 2N,. \displaystyle\Vert\mathcal{F}^{-1}[|$\xi$|^{-2+\frac{n}{p} $\xi$\cdot\int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)\^{u}0($\eta$)\frac{$\eta$}{|$\eta$|^{2}\frac{e^{-t|$\xi$|^{2}(e^{2t($\xi$-$\eta$)\cdot$\eta$}-1)}{2($\xi$-$\eta$)\cdot$\eta$}d$\eta$]\Vert_{Lp} \displaystyle\cong\Vert\mathcal{F}^{-1}[|$\xi$|^{-2+\frac{n}{p}|$\xi$|^{2}\int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)\^{u}0($\eta$)\frac{1}{|$\eta$|^{2}\frac{e^{-t|$\xi$|^{2}(e^{2t($\xi$-$\eta$)\cdot$\eta$}-1)}{2($\xi$-$\eta$)\cdot$\eta$}d$\eta$]\Vert_{L^{p}. (1.13). \cong \Vert \mathcal{F}^{-1}[| $\xi$|^{\frac{n}{p} $\chi$( $\xi$-2N)]\Vert_{L^{p} N^{2(2-\frac{n}{p})}N^{-2}N^{-2} \cong(2N)^{\frac{n}{p} N^{-\frac{2n}{p} , \cong N^{-\frac{n}{p}}. and the last term is bounded for all N \in N.. of the nonlinear term the. norm. as. u. 1. in the. give. a. large. case. remark that the. equation \partial_{t}u-\triangle u. =. p > 2n. one can. we can. expect the divergence. by (1.12). (1.13) although. and. expect the existence of solutions. and the initial datum is small.. Comparison. §2.. scaling. \rightarrow. of the initial data is bounded. Then. such that the solution is. We. N. Therefore,. with the nonlinear term u^{2}. optimal. space of. well‐posedness for. the nonlinear heat. u^{2} is different from that for the drift diffusion equation, where. invariant spaces. are common. and. \dot{B}_{p,q}^{-2+\frac{n}{p} (\mathb {R}^{n}) (1 \leq p, q \leq \infty). .. In. conclusion,.

(10) Remarks. shall. we. explain. that the critical space for the nonlinear term u^{2} is. for the drift diffusion equation to obtain the. since the term. well‐posedness. than that. worse. u^{2} is. not. a. form.. divergence Let. 39. on the 1LL‐Posedness results for the drift diffusion system. us. consider 2 space dimensions for. For the. simplicity.. following. heat. equation;. \left\{ begin{ar y}{l \parti l_{t}u-\triangleu= ^{2}&t>0,x\in\mathb {R}^{2},\ u(0,x)=u_{0}(x)&x\in\mathb {R}^{2}, \end{ar y}\right. it is. possible. to show the. H^{-1}(\mathbb{R}^{2}) corresponds. The space. \dot{B}_{p,q}^{-2+\frac{2}{p} (R2), and let the. through. us. analogous. explain. in the space. to the. case. the essential. p. H^{s}(\mathbb{R}^{2}). =. n. reason. 2 for the. =. for the. (1.12). Similarly. observation to. with s\leq -1. to. by the. scaling. (1.11),. [10].. critical space. of the. optimality. result. case. p=n. let u_{0} be defined. by. u_{0} :=N^{2-\frac{2}{p}}\mathcal{F}[ $\chi$(\cdot-Ne_{1})+ $\chi$(\cdot+N \mathrm{e}1)],. (2.1) and. ill‐posedness. we see. that. \displaystyle \sup_{N\geq 1}\Vert u_{0}\Vert_{H_{p}^{-2+\frac{2}{p} } <1. Here. note that the initial data. we. nent N and -N , and there is. estimate. (2.2). below.. Let. us. a. (2.1). is taken such that u_{0} have the. high‐high frequency. consider the solution. frequency. interaction to low u. satisfying. the. compo‐. frequency. for the. integral equation;. u(t)=e^{t\triangle}u_{0}+\displaystyle \int_{0}^{t}e^{(t- $\tau$)\triangle}u( $\tau$)^{2}d $\tau$. Approximating the low. the nonlinear term. by. u( $\tau$)^{2}\simeq(e^{ $\tau$\triangle}u_{0})^{2}. ,. we. have from the ristriction to. frequencey part by $\chi$( $\xi$). \displaystyle\Vert\int_{0}^{t}e^{(t-$\tau$)\triangle}(e^{$\tau$\triangle}u_{0})^{2}d$\tau$\Vert_{H^{-2+\frac{2}{p} (2.2). =\displaystyle\Vert\mathcal{F}^{-1}[(1+|$\xi$|)^{-2+\frac{2}{p} \int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)\^{u}0($\eta$)\frac{e^{-t|$\xi$|^{2} (e^{2t($\xi$-$\eta$)\cdot$\eta$}) {2($\xi$-$\eta$)\cdot$\eta$}d$\eta$]\Vert_{L^{p} \displaystyle\geq\Vert\mathcal{F}^{-1}[$\chi$($\xi$)\cdot1^{-2+\frac{2}{p} .\int_{\mathb {R}^{n} \^{u}0($\xi$-$\eta$)\^{u}0($\eta$)\frac{e^{-t|$\xi$|^{2} (e^{2t($\xi$-$\eta$)\cdot$\eta$}) {2($\xi$-$\eta$)\cdot$\eta$}d$\eta$]\Vert_{L^{p} \displaystyle \geq c(N^{2-\frac{2}{p} )^{2}\cdot\frac{1}{N^{2}. =cN^{2-\frac{4}{p}} \rightarrow 1. as. N\rightarrow 1. In the third line of the above. | $\eta$|. if. p>2.. estimate,. we. \simeq N obtained for the initial data u_{0}. used the restriction. given by (2.1).. By. | $\xi$|. this. \leq 1,. | $\xi$- $\eta$|. \simeq N and. observation,. one can.

(11) Tsukasa Iwabuchi. 40. and. expect that the problem is ill‐posed for the can. be. arbitrary large. for initial data which. Drift diffusion equations of. zero. structure. \dot{B}_{p,\infty}^{-2+\frac{n}{p} (Rn),. spaces. causes. p. embeddings. \geq are. and. scaling and. order,. (2.2) imply. equation one. are. order. common. that smaller order of. in the smaller spaces.. obtained with the indices p. monopolar type, p\geq 2n for. the. invariant spaces. the drift diffusion. for the heat equation with the nonlinear term. n. and heat. bipolar type. Indeed, 1. =. in Besov. for the drift. equation of bipolar u^{2} The following .. known. \dot{B}_{n,q}^{-1}(\mathb {R}^{n}) \subset\dot{B}_{2n,q}^{-\frac{3}{2} (\mathb {R}^{n}) \subset\dot{B}_{\infty,q}^{-2}(\mathb {R}^{n}) where. since the nonlinear term. form of the two. the. (1.12), (1.13) are. n). bounded.. divergence. ill‐posedness. ill‐posedness. diffusion equation of. type and. the. are. Although. order, respectively.. for all equations, the observations. divergence. p > 2 (=. case. monopolar type. with the nonlinear term u^{2} have the and the. Takayoshi Ogawa. 1\leq q\leq 1 Therefore, the .. does not have the. divergence. spaces of. well‐posedness for. ,. the. nonlinearity u^{2}. ,. which. form structure, should be smaller than those for the drift. diffusion equations.. Acknowledgments. Aid for. by. The work of T. Iwabuchi is. Young Scientists (B) #25800069.. JSPS Grant in Aid Scientific Research. partially supported by JSPS Grant. The work of T.. Ogawa. is. in. partially supported. (S) #25220702.. References. [1] [2] [3] [4] [5] [6] [7]. [8] [9]. Biler, J. Dolbeault, Long time behavior of solutions to Nernst‐Planck and Debye‐Hünkel drift‐diffusion systems, Ann. Henri Poincaré, 1 (2000), 461‐472. J. Bourgain, N. Pavlović, Ill‐posedness of the Navier‐Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), no. 9, 2233‐2247. P. Biler, M. Cannone, I. A. Guerra, G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693‐708. M. Cannone, F. Planchon, Self‐similar solutions for Navier‐Stokes equations in \mathbb{R}^{3} Comm. Partial Differential Equations 21 (1996), no. 1‐2, 179‐93. L. Corrias, B. Perthame, H. Zaag, Global solutions of some chemotaxis and angiogenesis system in high space dimensions, Milan J. Math., 72 (2004), 1‐28. W. Fang, K. Ito, Global solutions of the time‐dependent drift‐diffusion semiconductor equa‐ tions, J. Differential Equations, 123 (1995), 523‐566. H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983, in: Proc. Internat. Conf. on Wavelets and Applications (Radha, R.; Krishna, M.; Yhangavelu, S. eds New Delhi Allied Publishers, 2003, 1‐56. H. Gajewski, K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl., 113 (1986), 12‐35. T. Iwabuchi, Global well‐posedness for Keller‐Segel system in Besov type spaces, J. Math. Anal. Appl., 379 (2011), no. 2, 930‐948. P.. ,.

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