Browsing by Author "Miklavcic, Milan"

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    Non-linear Stability of Asymptotic Suction
    (1982) Miklavcic, Milan
    A flow over a plane y = 0 in R3 given by U(x,y,z) = (1 - e-y, -1/R, 0) is called an asymptotic suction velocity profile [12]. R>0 is the Reynolds number. U satisfies the Navier-Stokes equation ðv / ðt + (v · ) v = - p0 + 1/R v div v = 0 with p0 = 0. In the present paper it is proved that the stability of U for small perturbations which initially decay exponentially in the y direction and are periodic in the x and z direction is governed by the eigenvalues of the classical Orr-Sommerfeld equation [1, 8, 12]. For precise statements see Theorems 4, 5, 9, and 15.
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    Stability for Semilinear Parabolic Equations with Noninvertible Linear Operator
    (1983) Miklavcic, Milan
    Suppose that x'(t) + Ax(t) = f(t, x(t)), t ≥ 0 is a semilinear parabolic equation, e-At is bounded and f satisfies the usual continuity condition. If for some 0 < ≤ 1, 0 < < 1, p > 1, > 1 ||t Ae-At|| ≤ C, t ≥ 1 ||f(t, x)|| ≤ C(||A x|| p + (1 + t)-), t ≥ 0 whenever ||A x|| + ||x|| is small enough, then for small initial data there exist stable global solutions. Moreover, if the space is reflexive then their limit states exist. Some theorems that are useful for obtaining the above bounds and some examples are also presented.