We establish the existence and uniqueness of local strong solutions to the Navier-Stokes equations with arbitrary initial data and external forces in the homogeneous Besov-Morrey space. The local solutions can be extended globally in time provided the initial data and external forces are small. We adapt the method introduced in [23], where the Besov space is considered, to the setting of the homogeneous Besov-Morrey space.1 space Y to (1.1) is the so-called Serrin class L α (0, ∞; L p (R n )) for 2/α + n/p = 1 with n ≤ p ≤ ∞. For the initial data a and the external force f , the corresponding scaling laws are like a λ (x) = λa(λx) and f λ (x, t) = λ 3 f (λx, λ 2 t), respectively. Therefore, it is suitable to solve (1.1) in Banach spaces X for a and Y for f with the properties that a λ X = a X and f λ Y = f Y for all λ > 0, respectively.Let's first recall some results with respect to the space X. Since the pioneer work of Fujita-Kato [13], many efforts have been made to find such a space X as large as possible. For instance, Kato [17] and succeed to find the space X = L n (R n ). and Cannone-Planchon [9] whereḞ s q,r (R n ) denotes the homogeneous Triebel-Lizorkin space. The result in [19] seems to be optimal in the sense that continuous dependence of solutions with respect to the initial data breaks down in X =Ḃ −1 ∞,r (R n ) for 2 < r ≤ ∞, which was proved by Bourgain-Pavlović [5], Yoneda [35] and Wang [34]. Amann [4] has established a systematic treatment of strong solutions in various function spaces such as Lebesgue space L p (Ω), Bessel potential space H s,p (Ω), Besov space B s p,q (Ω) and Nikol'skii space N s,p (Ω) in general domains Ω. In this direction, based on the Littlewood-Paley decomposition, Cannone-Meyer [7] showed how to choose the Banach spaces X for a and Y for u. Besides these results, in terms of the Stokes operator, Farwig-Sohr [11] and Farwig-Sohr-Varnhorn [12] proved a necessary and sufficient condition on a such that weak solutions u belong to the Serrin class.On the other hand, in comparison with a number of papers on well-posedness with respect to the initial data, there is a little literature for investigating the suitable space Y of external forces f satisfying f λ Y = f Y for all λ > 0. For instance, proved existence of strong solutions forfor some δ > 0, where P denotes the Helmholtz projection. Cannone-Planchon [10] treated the case n = 3 and showed thatfor 2/α + 3/p = 2 with 2/3 < p < ∞ is a suitable space. See also Planchon [30]. After introducing the space of pseudo measures PM k = a ∈ S ′ ; sup ξ∈R n |ξ| k |â(ξ)| < ∞ , Cannone-Karch [8] showed that the pair of X = PM 2 and Y = C w (0, ∞); PM 0 is suitable for n = 3. Recently, Kozono-Shimizu [21] constructed mild solutions for X = L n,∞ (R n ) and Y = L α,∞ (0, ∞; L p,∞ (R n )), where 2/α + n/p = 3 and max{1, n/3} < p < ∞. Another choice of X and Y was obtained by Kozono-Shimizu [22] which proved the existence of mild solutions in the case when X =Ḃ −1+ n p p,∞ (R n )andfor n < p < ∞, n/3 < p 0 ≤ p and s 0 < min{0, n/p 0 − 1...

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In this paper, we investigate some special regularities and decay properties of solutions to the initial value problem(IVP) of the Benjamin equation. The main result shows that: for initial datum u0 ∈ H s (R) with s > 3/4, if the restriction of u0 belongs to H l ((x0, ∞)) for some l ∈ Z + and x0 ∈ R, then the restriction of the corresponding solution u(·, t) belongs to H l ((α, ∞)) for any α ∈ R and any t ∈ (0, T ). Consequently, this type of regularity travels with infinite speed to its left as time evolves.MSC: primary 35Q53, secondary 35B05.