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Regularity and uniqueness in quasilinear parabolic systems

Pavel Krejčí, Lucia Panizzi (2011)

Applications of Mathematics

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain ω Ω n , n { 2 , 3 } . The result that we obtained in this paper is as follows. Suppose that v ^ ( t , x ) is a given solution of the Navier-Stokes equations. Let v 0 ( x ) be a given initial condition and v ^ ( 0 , · ) - v 0 < ε where ε is small enough. Then there exists a locally distributed control u , supp u ( 0 , T ) × ω such that the solution v ( t , x ) of the Navier-Stokes equations: t v - Δ v + ( v , ) v = p + u + f , div v = 0 , v | Ω = 0 , v | t = 0 = v 0 coincides with...

Remarks on exact controllability for the Navier-Stokes equations

Oleg Yu. Imanuvilov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain ω Ω n , n { 2 , 3 } . The result that we obtained in this paper is as follows. Suppose that v ^ ( t , x ) is a given solution of the Navier-Stokes equations. Let v 0 ( x ) be a given initial condition and v ^ ( 0 , · ) - v 0 < ε where ε is small enough. Then there exists a locally distributed control u , supp u ( 0 , T ) × ω such that the solution v(t,x) of the Navier-Stokes equations: t v - Δ v + ( v , ) v = p + u + f , div v = 0 , v | Ω = 0 , v | t = 0 = v 0 coincides...

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