由修正的Navier─Stokes方程产生的动力系统（Ⅰ）

Dynamical Systems Generated by the Modified Navier-Stokes Equations

证明了由于１９６８年提出的修正的Ｎａｖｉｅｒ─Ｓｔｏｋｅｓ方程可产生一个作用于一可分的Ｈｉｌｂｅｒｔ空间Ｈ上的单参数半群｛Ｖ＿ｔ｝，算子Ｖ＿ｔ于ｔ＞０连续，于ｔ＞０紧。给出了｛Ｖ＿ｔ｝成为连续半群的充分条件。证明了此半群有非空的紧的不变的可吸引Ｈ中任何有界集的吸引子μ．对于Ｈ中任何有界不变集Ａ（可以是μ），从Ａ上出发的运动的决定模态的个数有限。如果Ａ是Ｈ中的紧不变集（可以是μ），则Ｖ＿ｔ（ｔ≥０）在Ａ上可逆从而容许定义Ｖ＿ｔ＝Ｖ￣（－１）＿ｔ（ｔ＜０）使得｛Ｖ＿ｔ，ｔ∈Ｒ，Ａ｝是一连续群，即在经典意义下的动力系统。证明了Ｈ中任何紧不变集的Ｈａｕｓ─ｄｏｒｆｆ维和分维有限。

It is proved from the Modified Navier-Stokes equations proposed by O.A.Ladyzhenskaya in1968 1t is possible to derive a single parameter semigroup {V_t,t≥0,H}on a separable Hilbertspace H, where the operator V_t is continuous for t≥0 and compact for t>0. A sufficient conditionis given for V_t to be a continuous semigroup.It is also proved that this semigroup has a nonemptyminimal compact invariant attractor μ which can attract any bounded set in H.For any boundedinvariant set A (may be identical with μ), the number of determining modes for any motionstarting from A is finite. It is also proved that the Hausdorff dimension and the fractal dimensionfor any compact invariant set in H (may be identical with μ) are both finite.