Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter τ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow \infty $$\end{document}. For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces B˙q,σs(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{B}^s_{q,\sigma }({\mathbb {R}}^n)$$\end{document} and F˙q,σs(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{F}^s_{q,\sigma }({\mathbb {R}}^n)$$\end{document} for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

where u τ = u τ (t, x) and ψ τ (t) = ψ τ (t, x) denotes the unknown density of molds and the distribution of the chemical substance, (u 0 , ψ 0 ) is given pair of initial data and λ ≥ 0 is a constant. The parameter τ > 0 denotes the relaxation time and when the dynamics of the chemical substance is relatively slow the zero relaxation time limit can be considered. By passing τ → ∞, the limiting functions formally solve the initial value problem of the drift-diffusion system: (1.3) The system (1.3) can be also found by a singular limit problem from the compressible Navier-Stokes equations [15,25]. The system (1.1) is called as the Patlak-Keller-Segel system (Patlak [52], Keller-Segel [24]), while (1.3) is called drift-diffusion system (Mock [37]) or the simplified Keller-Segel system (Jäger-Luckhaus [22], Biler [3], Nagai [38]). A large number of literature on the research of the well-posedness, the ill-posedness and asymptotic behaviors of the solutions have been appeared ( [3][4][5][6]8,10,11,[19][20][21][22][23][30][31][32][33][38][39][40][41][42][43]54,55,58,61] and references therein). The existence of solutions and the well-posedness of the problems (1.1) and (1.3) are known in the scaling invariant class, namely the critical scaling class by influence of results due to Serrin and Fujita-Kato on the incompressible Navier-Stokes equations (see [3,5,29,31,32]). In particular, the systems (1.1) and (1.3) are invariant under the following scaling transformation with the restriction λ = 0: For μ > 0, u μ (t, x) = μ 2 u(μ 2 t, μx), ψ μ (t, x) = ψ(μ 2 t, μx). (1.4) Then the invariant Bochner class is given by the following: (1.5) It is natural and optimal to choose θ = σ = ∞ in order to solve the problem (1.3) in the method of the heat semi-group, then we find that the invariant exponents are q = n 2 and r = ∞. However, by the elliptic and parabolic regularity theory, it is difficult to choose the solution space L ∞ (R n ) for the critical space of ψ τ and ψ and one of the corresponding regularity class is the bounded mean oscillation B M O(R n ) or the vanishing mean oscillation V M O(R n ). The equation (1.3) can be solved in such critical spaces and Kozono-Sugiyama-Yahagi [29] first pointed out that the Keller-Segel system (1.3) is globally well-posed in such a critical class involving B M O(R n ) for small initial data. Nagai-Ogawa [41] considered the class of B M O for constructing a global mass threshold solution in the two-dimensional drift-diffusion system (1.3) with λ = 0 (cf. Wei [60]).
In this paper, we consider the singular limit problem for the system (1.1) in the finite mass case. Raczyński [53] and Biler-Brandolese [4] studied the singular limit problem in two space dimensions. They considered the case of the vanishing initial condition ψ 0 ≡ 0 and showed the strong convergence (1.2) for a small data global solution in the scaling invariant spaces such as the class of pseudo-measures or the Lorentz spaces. Their topologies for convergence on lim τ →∞ ψ τ are not relevant of an initial layer since they posed a time weight function in the metric so that the initial behavior is neglected in the topology of convergence. Lemarié-Rieusset [36] also showed the singular limit in the scaling invariant space based on the Morrey space. In the previous works [34,35], the authors developed the singular limit problem with a presence of the initial layer on ψ τ (t) and showed the strong convergence in the standard space-time Lebesgue-Bochner space uniformly including a large data time-local solution.
Since we need to introduce a common function space between two problems (1.1) and (1.3), we introduce an admissible class for showing the singular limit [34,35] such as ∇ψ ∈ L θ R + ; L r (R n ) , 2 θ + n r = 1, n < r < ∞. (1.6) Let us introduce a common critical space to both the problems (1.1) and (1.3) and consider the singular limit problem (1.2).
Definition (Admissible pairs). For a pair of the exponents (θ, q), (σ, r ) they are called the scaling invariant (Serrin) admissible if ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2 θ + n q = 2, n 2 < q < θ, 2 ≤ θ, 2 σ + n r = 1, n < r < σ, 1 ≤ rq r + q . (1.7) We frequently choose θ = σ as in (1.6) and we regard (θ, q) and (θ, r ) as the admissbile in the following. Then the solution is invariant under the scaling in the Bochner spaces (1.7) above. The spatial exponents are restricted by (1.7) and is excluded in (1.7) in [34,35], where BC(I ; X ) denotes the class of bounded continuous functions from an interval I to a Banach space X , and such estimates are treated in a different way.
We first show the time local well-posedness and the global well-posedness with small data to both (1.1) and (1.3) in the scaling critical function space, i.e., in the Bochner space with the admissible exponents. Then we show the singular limit problem τ → ∞ for the solution to the Keller-Segel system (1.1) and the limiting function indeed solves the problem (1.3). To this end, we introduce a generalization of the space time estimate for the parabolic equation in the critical homogeneous Besov spaces and apply such estimates to establishing the singular limit problem in the scaling invariant class (1.4). We notice that it is possible to construct a solution to (1.1) in the class indicated in (1.7) with different exponents θ = σ , however this is not suitable for the singular limit problem, since the solution to the limiting problem (1.3) has to have the same exponent for the time integration. We also note that a slightly general system including the medical models involving the chemotaxis structure are considered and a very much related singular limit is also proven. See for instance [44]. The main difference of results here from the former results [34,35] is that the solution here is considered in the end-point critical function class V M O(R n ) for time local solution and B M O(R n ) for small solution. In the proof, generalized maximal regularity is extended in Theorem 2.5 (more precisely, Proposition 5.6) and it is used for that purpose. We also employ maximal regularity in the class of B M O(R n ) shown in Ogawa-Shimizu [48].

The local and global well-posedness
To consider the well-posedness of the system (1.1) in both the time local and the time global cases, one of difficulty lying in the problem is regularity for the limiting solutions ψ since it must solve either the Poisson or the Helmholtz equations and under the critical setting; u ∈ L n 2 (R n ), ψ does not necessarily belong to L ∞ (R n ), where L ∞ (R n ) is the corresponding scaling critical space observed in (1.5). To avoid such a difficulty, we introduce the class of bounded mean oscillation (B M O).
For a measurable function and B R (x) denotes n-dimensional ball centered at x with a radius R > 0. For s > 0, leṫ The classes B M O and V M O are quasi-Banach spaces and if we identify the elements of B M O up to constant, they are regarded as the Banach space. We also introduce the spacetime space L 2 (I ; B M O(R n )) that is essentially (but only inhomogeneous version) used by Koch-Tataru [26] for solving the incompressible Navier-Stokes equation in the limiting scaling invariant class. Definition. For I = (0, T ) with T ≤ ∞, The corresponding space L 2 (I ; V M O(R n )) is similarly introduced by applying the closure. The equivalent norm in the above can be identify from this definition of mean average immediately.
We aslo introduce the homogeneous Besov spaces (cf. [59]). Let {φ j } j∈Z be the Littlewood-Paley dyadic decomposition of unity, i.e.,φ is the Fourier transform of a smooth radial function φ with φ(ξ ) ≥ 0, for all ξ = 0. For s ∈ R and 1 ≤ p, σ ≤ ∞, we define the homogeneous Besov spacė with c n = (2π) −n/2 and the norm where f * g = c −1 n f * g denotes the convolution with a constant correction. We also define the inhomogeneous Besov space B s p,σ (R n ) by with sn the norm We define the mild (strong) solution of system (1.1) and (1.3). Let e tΔ denote the heat evolution operator given by is the Gauss kernel for t > 0.
with the Bessel potential given by and the heat kernel G t (x) = (4πt) −n/2 e −|x| 2 /4t . We choose pairs of exponents for the solution class as (θ, q) and (σ, r ) defined in (1.7). Then a natural class for the common initial data is θ = σ and it is indeed given by the sharp trace estimate from the semi-group representation in the real interpolation theory such as e tΔ u 0 L θ (I ;L q ) < ∞, Or one can restrict the class of ψ 0 itself by choosing at θ = 2 and r = ∞ to have (1.14) Hence we introduce a common class for the initial data and consider the equation in the class One can find regularity of the solution ψ in Proposition 1.2 as ψ ∈ B M O(R n ) (cf. [28,29] Then there exist T = T (u 0 , ψ 0 ) > 0 and the unique strong solution (u τ , ψ τ ) to (1.1) in Furthermore, the solution satisfies the regularity estimates: For any admissible pairs (θ, q) and (θ, r ), there exists M > 0 independent of τ > 0 and λ ≥ 0 such that r ,θ (R n ) and for some ε 0 > 0, where R + = (0, ∞). Furthermore, the solution satisfies the a priori estimate: For any admissible pairs (θ, q) and (θ, r ), The extra assumption (u 0 , ψ 0 ) ∈Ḃ r ,θ (R n ) on the initial data is required for estimates involving maximal regularity. Indeed, such an assumption on the initial data can be removed if we employ the weaker topology for constructing the solution such as sup t>0 u τ (t) Then the assumption on the initial data can be relaxed into the simplest way However those function spaces are not suitable for applying the singular limit problem directly as we see below (see Sect. 1.3). Then we modify the existence class as following: For any small η 0 > 0 This is possible because the solution is getting smoother after t > 0 and (u τ , ψ τ ) ∈ . The corresponding solvability of the initial value problem (1.3) has already known in non-critical space (Kurokiba-Ogawa [31,32]), and the critical space (Nagai-Ogawa [41]).
q,θ (R n ). If n = 2, then assume further that (1.17) Then there exists T = T (u 0 ) > 0 such that the unique strong solution (u, ψ) to (1.3) exists and Furthermore, the solution satisfies the a priori estimate: Remark. When n = 2, the extra-assumption on the initial data u 0 ∈Ḃ − 2 θ q,θ (R 2 ) with the admissible relation 1 θ in Theorem 1.1 and Proposition 1.2 prevents the non-negative data u 0 ≥ 0 unless it is trivial, since the regularity restriction necessarily implies the mean value zero of its elements (cf. [21]). In order to avoid such inconvenience, we assume that the initial data u 0 belongs to the In this case, all the statement remains valid over the time local interval I = (0, T ) for arbitrary T < ∞. However the usage of inhomogeneous Besov space does not allows to apply maximal regularity uniformly in time. The constant appears in the estimate of maximal regularity depends on T as C O((log T ) 1/θ ) as T → ∞ (see Appendix below for the detail). Hence the statements should be restricted for arbitrary finite interval and the bound in (1.15) depends on T .
As is well-known, the threshold mass M * in (1.18) is known as 8π if n = 2 and u 0 is non-negative [3,5,41,43]. For the large initial data u 0 1 > 8π, the positive solution blows up in a finite time (Bilar [3], Nagai [38,39], Kurokiba-Ogawa [31], Nagai-Ogawa [40], Wei [60]). If n ≥ 3 then the M * ≥ 8 However we expect that the threshold for the global existence may be where C H L S is the best possible constant of the Hardy-Littlewood-Sobolev inequality For the large data case, the positive solutions blow up in a finite time, namely there exists a maximal existence time 0 < T * < ∞ such that if the initial data satisfies where which stands for the Helmholtz free energy (see Calvez-Corrius-Ebdes [9], Ogawa-Wakui [51]). Indeed, one can observe that a solution has a concentration phenomena when the solution blows up in finite time (see [50]). We should like to notice that the initial data satisfies (1.20). The existence and the uniqueness of the solution to (1.3) for u 0 ∈ L n 2 (R n ) is considered in Kozono-Sugiyama-Yahagi [29], where ψ ∈ C(I ; B M O(R n )).
The main difference between the previous result [34] and ours is that ψ τ nor ψ do not belong toẆ 1,2 (R 2 ) when n = 2 with λ = 0. Therefore we avoid to choose function space L θ (0, T ;Ẇ 1,2 (R 2 )). To treat them in a unified way, we do not use the classẆ 1

Singular limit problem
The solution to (1.1) is formulated by the Duhamel formula as is defined in (1.11). One can find that the singular limit problem accompanies with the initial layer because of the presence of the initial data ψ 0 . Since the system (1.1) and (1.3) have the common structure in the equation for u τ and u, the main issue is how to formulate for the equation of ψ τ and ψ. Indeed, noticing compare the equations ψ τ and ψ. By employing analogous argument for proving the existence of solution via the construction mapping theorem, we show the difference of those equation converges to 0 as τ → ∞ except the initial layer. The following singular limit result is shown for the case n = 2 in Kurokiba-Ogawa [35] (cf. for a higher dimensional non-B M O result [34]). We generalized it for higher dimensional case: Theorem 1.3 For n ≥ 2, let (θ, q) and (θ, r ) be admissible pairs defined in (1.7) and assume For T = ∞, the corresponding regularity conditions on the initial data (u 0 , ψ 0 ), i.e., r ,θ (R n ) with the smallness (1.16) be assumed.
1. (The existence of the limit solution) Then for the same initial data u 0 , there exists a unique strong solution (u, ψ) to (1.3) as If T < ∞, then ψ ∈ BC(I ; V M O(R n )) ∩ L θ (I ;Ẇ 1,r (R n )) . (1.21) Besides, ψ τ (t) has the initial layer as τ → ∞. Namely it never converges to ψ = We should like to emphasize that the solution to drift-diffusion equation ) at most. Namely the class of the solutions ψ τ and ψ of two system are different from each other. Nevertheless, the convergence is shown in the topology C(I t 0 ; L 1 (R n ))×C(I t 0 ; V M O(R n )) except the initial layer, where I t 0 = (t 0 , ∞)∩I with t 0 > 0 because the solution u τ has a higher regularity over t ∈ (0, T ) which enable us to treat the solutions ψ τ and ψ are both in V M O(R n ).
Let us consider the formulation how to prove the singular limit problem (1.2). The external term of the ψ τ -equation in (1.1) can be regarded by changing (1.22) Then by using the dissipative estimates for the heat equation of u, we want to see that Such a formal computation can be justified by employing the similar argument to construct the solution. In particular, in order to justify the above procedure, one may introduce a typical metric induced from the norm such as where (θ, q) and (θ, r ) are the Serrin admissible given in (1.7). However such a choice of metric does not work well for the critical cases since the integrability conditions and the stability conditions are not consistent in the following estimate: For the first term of the right hand side of (1.22), where B( p, q) denotes the Beta function given by Now we notice that the condition on the convergence of the Beta function requires while the stable condition on the exponent τ should be given by to justify the singular limit τ → ∞. Unfortunately those conditions do not hold simultaneously.

Usage of generalized maximal regularity
To overcome such a difficulty, we employ maximal or generalized maximal regularity for the heat equation and use the space time estimates. In general, for the solution of the Cauchy problem of the heat equation; maximal regularity is given by the following inequality: For any 1 ≤ ρ ≤ θ with (θ, q) as Serrin's admissible exponent, it holds that denotes the homogeneous Besov space given by (1.9). Applying such an estimate to the difference of the solutions to (1.1) and (1.3); ψ τ (t)−ψ(t). Since the difference solves the following equation: we apply the estimate (1.24) to the corresponding integral equation to the above equation with θ = 2 and q = ∞ for the main part I 1 that for all τ ≥ 2, where we used the Sobolev embedding L q (R n ) ⊂Ẇ −1,r (R n ) for 1/r = 1/q − 1/n. Hence we can avoid the difficulty appeared in (1.23) and can show the singular limit.
In what follows, for 1 ≤ p, r , θ ≤ ∞ and s ∈ R, let s p be a real sequence space determined by and let L p = L p (R n ) andẆ s, p =Ẇ s, p (R n ) be the Lebesgue and the homogeneous Sobolev spaces in the variable x whose norm is written by f p ≡ f L p and f Ẇ s, p ≡ |∇| s f p . Let L θ (I ; X ) be a Bochner class on the Banach space X over the time interval I = (0, T ) (T ≤ ∞). For s ∈ R and 1 ≤ σ ≤ ∞, letḞ s p,σ =Ḟ s p,σ (R n ) the homogeneous Lizorkin-Triebel spaces and the norms of the space is given by the following: For 1 ≤ p, σ ≤ ∞ and s ∈ R, where {φ j } denotes the Littlewood-Paley dyadic decomposition of unity and φ j * f denotes the convolution operation with φ j and f with a correction of a constant. In particular, we notice thatḞ s p,2 Ẇ s, p by the well-known Littlewood-Paley theorem. We also denote the homogeneous Besov spacesḂ s p,σ (R n ) asḂ s p,σ for simplicity. We should notice that the inclusion of the sequence space θ ⊂ σ directly gives thatḂ s for a test function ϕ ∈ S with ϕ λ (x) = λ n ϕ(λx) for λ > 0, and the Hardy-Sobolev spaceṡ H s,1 is analogously introduced in a similar way aṡ is the Banach spaces if s < n.

Inequalities and embeddings in the critical spaces
In general, the function inequality in two dimensional Euclidean space is different from higher dimensions. Here we summarized the Sobolev type inequalities in two dimensions.
Lemma 2.1 Let n ≥ 2 and let f = f (x) be a measurable function on R n . There exists a constant C = C(n) > 0 such that the following inequality hold: Proof of Lemma 2. 1 The inequality (2.1) is due to Gagliardo and Nash, and is obtained by a straightforward computation: Integrating the both side of the following identity and applying the Hölder inequality in each variable by n times, we obtain (2.1) (see e.g., Brezis [7]). The embedding (2.2) and (2.7) are direct consequences of the Bernstein type lemma and Hausdorf-Young's inequality. (2.3) and (2.4) follow from (2.1) and the boundedness of the singular integral operator in L n n−1 (R n ). The inequality (2.5) follows from the Poincaré inequality in n-dimensions: The inequality (2.6) is a consequence from (2.5) and the boundedness of the singular integral operators in B M O.
We notice that the following inequalities generally fail: (2.8)

Proof of Lemma 2.2 For f ∈ B M O(R n ), it is known that
(cf. Frazier-Jawerth [16]) and the inequality (2.8) follows immediately from (2.9) that sup x∈R n ,k∈Z Then the following continuous embeddings hold:

Proof of Lemma 2.3 Noticing the relations
the first embedding is due to the Sobolev inequality The second relation is due to the well-known theorem by Littlewood-Paley:Ḟ 0 q,2 (R n ) L q (R n ) for any 1 < q < ∞ (see Stein [56]). The third embedding is due to the property of the sequence spaces 2 ⊂ θ under 2 ≤ θ . The last embedding is given by the Minkowski inequality such as under the restriction on q and θ as q ≤ θ < ∞.

The heat evolution on VMO
It is well-known that the heat kernel has a dissipative estimate of L p -L q type: Here is a dissipative estimate for the heat kernel on B M O and V M O.

Generalized maximal regularity
We consider the Cauchy problem of the heat equation: For μ > 0, Then maximal regularity is given by the following way [12,46]: For any 1 ≤ ρ, p ≤ ∞, The first estimate is well-known result from the general framework [2,13,14,18] and the cases ρ = 1, ∞ are generally excluded since such spaces are not UMD (unconditional Martingale difference) and it is not covered by the general theory of UMD. The remarkable feature of the latter estimates is that the estimate (2.16) allows the case ρ = ∞ and it is useful to estimate for applying the integral equation. On the other hand the latter estimate involves the homogeneous Besov spaces and it is not easy to make clear the relation between the Legesgue spaces sincė We state the following slightly general form of maximal regularity.

2.19)
The analogous estimate of (2.19) between the inhomogeneous Besov spaces also holds with the bound C = C T is depending on T < ∞.
The proof of Theorem 2.5 (1) (2) are given in [34] (cf. [47]). For the completeness, we show the out-lined proof in Appendix. The proof of (2.19) is in Proposition 5.6 in Appendix, too.
We also show the maximal regularity estimate in B M O in the composite space-time space L 2 (I ; B M O(R n )) which is essentially due to Stein [57, p. 161] and explicitly appeared in Koch-Tataru [26] and Ogawa-Shimizu [48].
We consider maximal regularity for (2.15) in non-UMD Besov spaces (cf. Ogawa-Shimizu [48]). Koch-Tataru [26] obtained the homogeneous estimate in B M O by using the Carleson measure: Proposition 2.6 [26] Let e tΔ be the heat kernel and u 0 ∈ B M O(R n ). Then We show a sketch of alternative proof in Sect. 1 of Appendix below (see [26]).

Lemma 2.7
Let I = (0, T ) for some T > 0 and e tΔ be the heat semigroup for t > 0. Then It follows that Fixing n and passing |I | → 0 and applying the estimate (2.24), we obtain the result.
The following estimates are related to maximal regularity for the Cauchy problem of the heat equation: We consider the Cauchy problem of the heat equation: For μ > 0 and λ ≥ 0, The estimate for the homogeneous equation (i.e. the case for f ≡ 0 and λ = 0) is shown by Koch-Tataru [26] and applied to the well-posendess issue for the incompressible Navier-Stokes equations. As a generalization of the homogeneous estimate, maximal regularity for the Cauchy problem of the heat equation can be shown in Ogawa-Shimizu [48] for λ = 0 including its optimality of trace estimate (cf. Iwabuchi-Nakamura [20]).
that satisfies the following estimate: There exists a constant C > 0 independent of T , μ and λ such that . (2.26) The corresponding estimate in L 2 (I ; V M O(R n )) to (2.26) also holds under suitable conditions on u 0 and f .
The proof of Theorem 2.8 can be seen in [48]. For completeness, we show the out-lined proof of Theorem 2.8 in Appendix below.

Proof of local and global well-posedness
In this section, we show the local and global well-posedness of the solution to (1.1). The proof of Proposition 1.2 for the limiting equation (1.3) is indeed simpler than the proof of Theorem 1.1 and we do not show the proof for Proposition 1.2 (cf. [29,32]).
Proof of Theorem 1. 1 We first show the local existence of solution for I = (0, T ). Consider the mild solution to the corresponding integral equation: (3.1) Step 1 (The local well-posedness): Let τ > 0. We show the local in time existence and well-posedness of the solutions for the large initial data q) and (θ, r ) satisfy the Serrin admissible conditions with additional assumptions as follows: and set I = (0, T ) for some 0 < T < ∞ chosen later and let and N > 0 will be determined later. Introducing the metric on X T by we can show that X M is a complete metric space. Indeed By this fact, any Cauchy sequences converges a limit (u, ψ) in L θ (I ; L q ) × L θ (I ;Ẇ 1,r ) and weak- * compactness ensures that the limit indeed belongs to X T .
We then introduce a pair of the solution maps (Φ[u τ , ψ τ ], Ψ [u τ , ψ τ ]) as follows: For (3.4) and claim that the map Φ, Ψ is contraction in the critical space X T .
First we claim that (Φ, Ψ ) is onto X T . By maximal regularity (2.17) in Theorem 2.5 with s = −2, σ = 1 with the embeddings B 0 q,1 ⊂ L q to see that and by Proposition 2.6, It also follows for ψ 0 that (3.7) From (2.21) in Proposition 2.6 that for ψ 0 ∈ V M O, Hence from (3.5)-(3.8) and noting Lemma 2.7, we may choose the time interval |I | ≤ T sufficiently small such that for some small ε 0 > 0, 1 for any τ > 1 and the choice of T is independent of τ > 1. Let (θ, q) and (θ, r ) be admissible pairs with 1 < q < 2 < θ. We apply the bound for the initial data  1 The choice of T is independent of τ > 1.
for any 2 < θ and 1 < q < θ, and for some small ε 0 , C N 2 ≤ 1 2 N . By adding the both sides of the following embedding is realized: (3.11) We proceed the estimate for the map Ψ . From Theorem 2.8, (3.9) and (3.11), From (3.11) and (3.13), and choosing N > 0 small enough so that for some small ε 0 > 0, On the other hand, since the heat kernel is a bounded operator in V M O(R n ), we again use generalized maximal regularity (2.18), Lemma 2.2 and (3.10) to see that Hence we obtain from (3.15) that (3.17) Analogously from (3.10) and (3.17), we have Choosing ε 0 smaller as  . Let (u τ , ψ τ ) and (ũ τ , ψ τ ) be a solution of (1.1) corresponding to the initial data (u 0 , ψ 0 ) and (ũ 0 , ψ 0 ), respectively. Then very much similar estimate of (3.17) and (3.18), we obtain (3.21) From Theorem 2.8, is a continuous function on L n 2 , u τ ∈ BC I ; L n 2 (R n ) . Besides let (u τ , ψ τ ) and (ũ τ , ψ τ ) are two solutions of (1.1) corresponding the initial data (u 0 , ψ 0 ) and (ũ 0 , ψ 0 ), respectively. Then,  Step 2 (Global existence for small data). Since our function space in the previous step is scaling invariant, the global existence for (1.1) also follows almost similar (but somewhat simpler) way to the case for local well-posedness. Let (θ, q), (θ, r ) be admissible given by (3.2). We also call (θ, q) ∈ ∞, n 2 , (2, n) , (θ,r ) ∈ (2, ∞), (∞, n) the end-point admissible pairs. Fixing the admissible pair for I = R + as (θ, q), (θ, r ), we introduce the complete metric space: , u L θ (I ;L q ) , is chosen small later. For any admissible exponents (θ, q) and (σ, r ) (not the end-point exponents), we define the metric on X ε 0 by By this metric, X ε 0 is a complete metric space. For under C ε 0 ≤ 1 2 . All the above estimate in (3.25) is true for a finite time interval and we may prolong such an estimate for all time.
For any admissible exponents (θ, q) and (θ, r ), for C ε 0 ≤ 1 2 . In particular for the end-point admissible case (θ, q) = (2, n), by the same choice of the embedding classes, we see that Here we need one more restriction rq q+r ≤ 2 and by (1. for 2C ε 0 ≤ 1 2 . From (3.29) and (3.31), Combining (3.28) and (3.32), we obtain that Φ[u τ , ψ τ ], Ψ [u τ , ψ τ ] ∈ X ε 0 . Analogously from (3.17) and (3.18) for the difference of solutions (3.33) Choosing M smaller as if necessary, we have from (3.33) that under the smallness assumption on the initial data. Thus the map (Φ, Ψ [Φ]) is contraction onto X ε 0 and the Banach fixed point theorem implies that there exists a unique fixed point (u τ , ψ τ ) ∈ X ε 0 that solves the Eq. (1.1) in the critical space. In particular, from (3.28) and (3.32), the a priori estimate holds: where the constant M does not depend on the parameter τ > 0. For n = 2 we need a similar modification to the local well-posedness theory.

Proof for the singular limit
Proof of Theorem 1. 3 We first show Theorem 1.3 for the small data case: In what follows we assume that λ = 0 for simplicity. For the other cases λ > 0, all the proof can be applicable in the same way, since the evolution e τ t(Δ−λ) satisfies the same estimate with the case λ = 0.
Step 1 : For any η 0 > 0, let I = (0, ∞) and we consider the difference of solutions between (3.1) and (1.12) as the following: For any t ∈ I , Choose the admissible exponents (θ, q) (θ, r ) such as Let the time interval I = (η 0 τ −1 , ∞) for any η 0 > 0. Introducing the difference of solution as we apply the similar estimate in (3.17), it follows from Lemma 2.3 that  We have from the second equation of (4.1), (2.18) in Theorem 2.5 and (θ, r ) is the admissible, i.e., 2 < r ≤ θ that for all τ ≥ 2, where C > 0 is independent of τ > 0. The third term of (4.3), we apply the Sobolev inequality and the mean value theorem to see that For treating the first term of the right hand side of (4.6), we proceed by changing the variable r = τ s (4.7) Then applying (2.18) in Theorem 2.5, we see by regarding μ → τ , that Similarly analogous estimate (3.26), it follows that .
). Let {φ j } j∈Z be the Littlewood-Paley decomposition of unity and u 0 ∈Ḃ 0 p,ρ . Then there exists a constant C > 0 such that for any k, k ∈ Z with k < k, holds, where the constant C > 1 is depending on n, p and φ 0 .
To prove Lemma 5.3, we invoke the following estimate (see [47,Lemma 3.4

Proof of Lemma 5.3 Then we see that
The following proposition is the key to proving Theorem 2.5.
Finally we treat the case ρ = ∞. In this case, we modify the above argument in the different interpolation parameter. Namely, we have from (5.8) that It follows by letting 1 μ + 1 ν + 1 = 2 and using the Hausdorf-Young inequality with 1 By using the fact that 2 j <λ≤2 j+1 dλ λ = R + χ j (λ) dλ λ = log 2, we apply the Hölder and the Minkowski inequalities for j to obtain and we obtain the desired estimate (5.10) by letting ρ = ∞ and the duality argument.
If we trade off the range of exponents to slightly restricted conditions, then we may generalize Proposition 5.5 which stems from an estimate appeared in Kozono-Ogawa-Taniuchi [27] and Ogawa-Suguro [49] for the Morrey spaces. Proposition 5.6 Let 1 ≤ p ≤ ∞, 1 < ν < ρ ≤ ∞, s ∈ R, 0 < r < 2 and {e tΔ } t≥0 be the heat semi group in R n . For any 0 < T ≤ ∞ let I = (0, T ). Then there exists a constants C > 0 independent of T such that (5.10) In the case of the inhomogeneous Besovs, the corresponding estimate holds with a constant C depending on T .

Proof of Proposition 5.6
Limiting the integral range 1 < ρ < ∞, we employ the straight forward argument to show (5.10). For any f ∈ L ν (I ;Ḃ 0 p,∞ ), let χ I (t) be the characteristic function of I = (0, T ). Then we apply the dissipative estimate for the heat kernel for all t > 0, where 0 < r < 2 (see [27,Lemma 2.2]) and the Hausdorf-Young inequality to obtain where 1 < ν < ρ < ∞, 0 < r < 2 and and it concludes to have where C is independent of T .
All the above estimates are valid even if we exchange the homogeneous Besov spaces into the inhomogenenous Besov spaces except the bound constant. In such cases, the dissipative estimate (5.11) can be modified by and the estimate (5.15) still remains valid if one modifies as follows: (5.16) and we obtain by setting C T ≡ C max j≤ j 0 and we conclude to have where C T C2 j 0 (1−1/ν) T 1−1/ν .

A.1 Proof for the Koch-Tataru estimate
In this subsection, we illustrate for completeness that another proof of Proposition 2.6 under a slightly general setting.

Proof of Proposition 2.6
Let R > 0 and be a smooth cut-off function. Integrate by parts, we have Integrate the both sides of (5.17) in t ∈ (0, R 2 ). Then for x ∈ supp ∇η R (x − x 0 ) and and for any x ∈ R n , ∇e tΔ u 0 (x) − ∇ y e tΔ u 0 (y) 2 dxdydt The last inequality follows from Fefferman-Stein's H 1 -B M O-duality (see [17]) and regarding the heat kernel as the test function in H 1 , Namely, it holds that ∇e tΔ u 0 (x) − ∇ y e tΔ u 0 (y) 2 dxdydt . A direct cosequence of Proposition 2.6 is the following estimate (see for the proof, Ogawa-Shimizu [48]): Proposition 5.7 [48] Let e tΔ be the heat kernel and u 0 ∈ B M O(R n ). Then sup x 0 ,R>0 , (5.23) where C 0 is independent of T > 0. The corresponding estimate in L 2 (I ; V M O) to (2.21) also holds.

A.2 Maximal regularity in BMO
Proposition 2.6 implies regularity estimate for the Laplacian. We extend this estimate to full maximal regularity to the initial value problem for the heat equation in B M O as in Theorem 2.8.

Proof of Theorem 2.8
The homogeneous estimate for the initial value problem with f ≡ 0 have seen in Proposition 2.6 and it suffices to show that the inhomogeneous estimate with u 0 ≡ 0. Let T ≤ ∞ and I = (0, T ) and f ∈ L 2 (I ; S) and g ∈ L 2 (I ; S). We show the inhomogeneous estimate in V M O(R n )-based space. The general case can be shown by applying the Hahn-Banach extension theorem (cf. [7]). For R > 0, let be a smooth cut-off function around x 0 ∈ R n and for λ ≥ 0 set Then d dt The first term of the right hand side of (5.26) can be treated by noticing ∇ x v(t, y)ds = ∇ y v(t, x)ds = 0 and integrate by parts of each derivative for x and y variables, respectively to see where we note that the last term in the right hand side of (5.27) is obtained from the one line above by symmetry of x-y exchanging. Combining (5.26) and (5.27) and using η R (x)∇η R (x) = ∇η R (x) we integrate it over t ∈ (0, R 2 ) to see (5.28) J 1 can be cancelled with the left hand side by choosing ε > 0 small. Let η R (x) be a smooth cut-off function whose support is a little wider than the support of the ball B R (x 0 ), (5.29) Noticing that |∇η R | ≤ C R J 2 is estimated as Here the integral can be split into r integration and s integration, the right hand side is expressed by a multiple integraion and using the Cauchy-Schwartz inequality and η R (x) ≤ χ B R (x 0 ) (x), 2 −n |B R | ≤ η R 1 to have the following; .
Here we have used the estimate in Proposition 5.7.
On the other hand, let the Riesz operator defined by R j f = F −1 ξ j i|ξ |f . Then we can regard (t, x) and

y)dxdy
≡ I I I 1 + I I I 2 . (5.31) Integrating the both sides of (5.31) over t ∈ (0, R 2 ), The first term L 1 can be cancelled by choosing ε > 0 small enough. . (5.33) Using |∇η R | ≤ C R , we estimate the third term L 3 by changing the order of the time integration to see . Combining Proposition 2.6, we conclude the desired estimate.