zbMATH — the first resource for mathematics

On the suitable weak solutions to the Navier-Stokes equations in the whole space. (English) Zbl 0615.35067
The author obtains the suitable weak solutions in the whole space \(\Omega ={\mathbb{R}}^ 3\) to the following Navier-Stokes system of equations describing the motion of a viscous incompressible fluid: \[ (1)\quad \partial u/\partial t+u\cdot \nabla u-\Delta u=f-\nabla p\quad in\quad Q_ T;\quad \nabla \cdot u=0\quad in\quad Q_ T,\quad u|_{t=0}=u_ 0(x), \] where the symbols have their usual meaning and \(Q_ T=]0,T[\times {\mathbb{R}}^ 3.\)
At first the author mentions that a suitable weak solution of the system (1) is \[ (2)\quad u\in L^ 2(0,T;V)\cap C_{deb}(0,T;H) \] which satisfies the local energy inequality \[ (3)\quad | u(t)|^ 2+\int^{t}_{t_ 0} | \nabla u|^ 2 d\zeta \leq | u(t_ 0)|^ 2+\int^{t}_{t_ 0} <f,u> d\zeta \] for almost all \(t_ 0\in [0,T]\) (included \(t_ 0=0)\) and for all \(t\geq t_ 0.\)
After this he constructs another class of suitable weak solutions utilizing (2) and (3) and verifying the local energy estimate \[ \int_{\Omega} | u(t)|^ 2 \phi +2\iint_{Q_ t} | \nabla u|^ 2 \phi \leq \int_{\Omega} | u_ 0(t)|^ 2 \phi + \]
\[ +\iint_{Q_ t} | u|^ 2(\frac{\partial \phi}{\partial t}+\Delta \phi)+\iint_{Q_ t} (| u|^ 2+2p)u\cdot \nabla \phi +\iint_{Q_ t} 2f\cdot u\phi \] for every regular non-negative scalar function \(\phi\) with compact support with respect to the space variables. \(L^ 2(0,T;V)\) is the Banach space of strongly measurable functions in ]0,T[ with values in V, \(C_{deb}(0,T;H)\) is the subspace of \(L^{\infty}(0,T;H)\) of functions u(t) which are weakly continuous on [0,T] with values in H where \(H=\{u\in {\mathbb{L}}^ 2\); \(\nabla \cdot u=0\}\) and \(V=\{u\in {\mathbb{H}}'\); \(\nabla \cdot u=0\}\) etc.
Reviewer: R.N.Pandey

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65Z05 Applications to the sciences
46E15 Banach spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
35K35 Initial-boundary value problems for higher-order parabolic equations
35A35 Theoretical approximation in context of PDEs
PDF BibTeX Cite