Summary
We consider the motion of a barotropic compressible fluid in a one dimensional bounded region with impermeable boundary, see equation (1.1). Here, u(t, q) denotes the velocity and v(t, q) the specific volume. The quantity log v(t, q) measures the displacement of v(t, q) with respect to the equilibrium v ≡ 1. For the sake of brevity we denote here different norms by the simbol ‖ ‖. We show that there is a positive constant r0=r0(μ), a small ball B1 (r) (with radius R1 (r),\(\mathop {\lim }\limits_{r \to 0} R_1 (r) = 0\)), and a large ball B(r) (with radius R(r),\(\mathop {\lim }\limits_{r \to 0} R(r) = + \infty \)) such that the following holds, for each r ε [0, r0 [(i) If ‖f(t)‖ < r for all t ≧ 0, and if ‖(u(0), log v(0))‖≦R(r) (i.e. (u(0), log v(0)) ε B(r)) then, for sufficiently large values of t, ‖(u(t), log v(t))‖≦R1 (r); (ii) The solutions starting at time t=0 from the large ball B(r) have all the same asymptotic behaviour (see (1.11)); (iii) If f is T-periodic then there is a (unique) T-periodic solution (u(t), log v(t)) inside the small ball B1 (r). This periodic solution atracts all solutions which intersect the large ball B(r). Periodic solutions had been previously studied only for very specific pressure laws, namely p(v)∼-log v and p(v)∼-v−1.
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da Veiga, H.B. Attracting properties for one dimensional flows of a general barotropic viscous fluid. Periodic flows. Annali di Matematica 161, 153–165 (1992). https://doi.org/10.1007/BF01759636
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DOI: https://doi.org/10.1007/BF01759636