On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity

Abstract

The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of B 1 ∞,1 . It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.

Metrical almost periodicity and applications

“…∞,1 (R n ) denotes the usual Besov space (see e.g., [26,Definition 2.1]); let the space B 0 ∞,1 (R n ) n be defined in the same way. It is well known that O. Sawada and R. Takada have proved, in [26,Theorem 1.5], that the almost periodicity of function u 0 (x) in R n implies that the solution u(•, t) of (1.2) is almost periodic in R n for all t ∈ [0, T ].…”

“…∞,1 (R n ) denotes the usual Besov space (see e.g., [26,Definition 2.1]); let the space B 0 ∞,1 (R n ) n be defined in the same way. It is well known that O. Sawada and R. Takada have proved, in [26,Theorem 1.5], that the almost periodicity of function u 0 (x) in R n implies that the solution u(•, t) of (1.2) is almost periodic in R n for all t ∈ [0, T ]. In [18,Example 8.1.4], we have analyzed the situation in which R is an arbitrary collection of sequences in R n , and u 0 (•) has the property that for each sequence (b k ) in R there exists a subsequence (b k l ) of (b k ) such that the sequence of translations (u…”

Spatially almost periodic solutions for active scalar equations

“…We recall that a function is almost periodic (in the sense of Bochner) if is relatively compact in , where denotes the space of bounded uniformly continuous functions on . Flows expressed by almost periodic functions are not only physically but also mathematically interesting (see previous studies ()). F M still includes various almost periodic functions not necessarily periodic.…”

Spatially almost periodic complex‐valued solutions of the Boussinesq equations

“…These classes of almost periodic functions play an important role in differential equations. The almost periodic case is very different from periodic case and the case of decaying at the space infinity see [13][14][15][16][17] and the references therein.Definition 1.4. For S ∈ N 0 = N ∪ {0}, we define…”