Existence of a strong solution of the Navier-Stokes equations
The author declares of solution of the sixth Millennium Prize Problem. Namely, let \(\Omega=(-\pi,\pi)^3\) the 3-D cube, \(Q_T=\Omega\times(0,T)\), the functions \(v(x,t)=(v_1,v_2,v_3)\) and \(p(x,t)\) satisfy the Navier-Stokes equations NEWLINE\[NEWLINE \frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=f,\quad \text{div}\,v=0,\quad x\in \Omega,\;t\in(0,T) NEWLINE\]NEWLINE with the initial condition NEWLINE\[NEWLINE v(x,0)=0,\quad x\in \Omega NEWLINE\]NEWLINE and periodic boundary conditions NEWLINE\[NEWLINE v(x,t)|_{x_k=-\pi}=v(x,t)|_{x_k=\pi},\quad p(x,t)|_{x_k=-\pi}=p(x,t)|_{x_k=\pi}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE\left.\frac{\partial v}{\partial x_k}\right| _{x_k=-\pi}=\left.\frac{\partial v}{\partial x_k}\right| _{x_k=\pi},\quad k=1,2,3,\quad t\in[0,T]. NEWLINE\]NEWLINE It is asserted that the problem has a unique strong solution \((v,p)\) for any \(f\in L_2(Q_T)\). This solution belongs to the class NEWLINE\[NEWLINE \frac{\partial v}{\partial t},\, \Delta v,\, (v\cdot\nabla v)v,\, \nabla p\in L_2(Q_T). NEWLINE\]NEWLINE The proof is based on the reducing of the initial-boundary value problem of the Navier-Stokes system to the initial problem for the operator equation NEWLINE\[NEWLINE u'(t)+Au+B(u,u)=f(t),\quad t\in(0,T),\quad u(0)=0, NEWLINE\]NEWLINE in a certain Hilbert space. Here \(A\) is a linear self-adjoint operator \(-\Delta+I\) and \(B(\cdot,\cdot)\) is a bilinear operator corresponding to the convective term. The solvability of the operator equation is deduced from a strong a priori estimate.NEWLINENEWLINEReviewer's remark: There were many reports of a strong solvability to the Navier-Stokes equations during the last fifty years. All of them were erroneous. It is necessary to take the result of the present paper critically and verify the proofs carefully.
scientific article; zbMATH DE number 6247768