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Register a new userClassification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains
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We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=left( begin{array}{c} varphi_1(r)v_1(theta) varphi_2(r)v_2(theta) end{array} right) ,$$ where $x=rcostheta$ and $y=rsintheta$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are H{o}lder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.
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In this paper, we investigate the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a helically symmetric spatial domain. When data is assumed to be helical invariant and satisfies the compatibility condition, we prove this problem has at least one helical invariant solution.
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We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
A hyperbolic relaxation of the classical Navier-Stokes problem in 2D bounded domain with Dirichlet boundary conditions is considered. It is proved that this relaxed problem possesses a global strong solution if the relaxation parameter is small and the appropriate norm of the initial data is not very large. Moreover, the dissipativity of such solutions is established and the singular limit as the relaxation parameter tends to zero is studied
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