In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins-Daruud et al. in [25]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction ϕ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.
In this paper, we establish regularity and uniqueness for solutions to density dependent nematic liquid crystals systems. The results presented extend the regularity and uniqueness for constant density liquid crystals systems, obtained by Lin and Liu (1995 Commun. Pure Appl. Math. XLVIII 501-37).
Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier-Stokes equations, is a mathematical analog of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier-Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.
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