Stabilization of the Simplest Normal Parabolic Equation
نویسندگان
چکیده
The simplest parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given. Introduction. This paper is devoted to the proof of stabilization of the simplest normal parabolic equation by starting control. Parabolic equations of normal type were introduced in [7]-[10] with the purpose to understand better how to solve hydrodynamics equations whose solutions does not satisfy inequality of energy type. 1 Let explain this a little bit more detailed. Energy inequality that is true for solutions v of 3D Navier-Stokes equations gives possibility to prove solvability of these equations in the class of weak solutions. In order to prove existence of strong solution for 3D Navier-Stokes equations it would be enough to establish solvability of 3D Helmholtz equations in class of weak solutions. 2 But since solutions of 3D Helmholtz system do not satisfy energy estimate, existence of weak solutions for these equations is not established until now despite of very big importance of this problem. The reason of nonfulfillment of energy inequality for solution of Helmholtz equations in 3D case is that image B(ω) of nonlinear operator B from Helmholtz equation is not orthogonal to the vector ω (i.e., besides orthogonal component vector B(ω) contains component Φω collinear to vector ω ). 3 If we change in 3D Helmholtz equations their nonlinear operator B(ω) on its component 2010 Mathematics Subject Classification. Primary: 37D10, 40H05; Secondary: 35K57, 35B38.
منابع مشابه
The geometric properties of a degenerate parabolic equation with periodic source term
In this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the Cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. Our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete Riemannian manifold floating in space RN+1and is tangent to the space RN at ∂H0(t); (2...
متن کاملNormal forms for parabolic Monge-Ampère equations
We find normal forms for parabolic Monge-Ampère equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exhaust the different classes of parabolic Monge-Ampère equations with symmetry p...
متن کاملSimplest Equation Method for nonlinear solitary waves in Thomas- Fermi plasmas
The Thomas-Fermi (TF) equation has proved to beuseful for the treatment of many physical phenomena. In this pa-per, the traveling wave solutions of the KdV equation is investi-gated by the simplest equation method. Also, the effect of differentparameters on these solitary waves is considered. The numericalresults is conformed the good accuracy of presented method.
متن کاملکاربرد روش معادله سهموی در تحلیل مسائل انتشار امواج داخل ساختمان
With the rapid growth of indoor wireless communication systems, the need to accurately model radio wave propagation inside the building environments has increased. Many site-specific methods have been proposed for modeling indoor radio channels. Among these methods, the ray tracing algorithm and the finite-difference time domain (FDTD) method are the most popular ones. The ray tracing approach ...
متن کامل