On the convergence at infinity of solutions with finite dirichlet integral to the exterior dirichlet problem for the steady plane Navier-Stokes system of equations

1988 ◽  
pp. 256-274
Author(s):  
Dan Socolescu
Keyword(s):  
Dirichlet Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
System Of Equations ◽  
Dirichlet Integral ◽  
Steady Plane

scholarly journals Suppressing blow-up by gradient-dependent flux limitation in a planar Keller–Segel–Navier–Stokes system

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Classical Solution ◽  
Smooth Function ◽  
Stokes System ◽  
Blow Up ◽  
Navier Stokes ◽  
Diffusive Fluxes ◽  
Fluid Interaction ◽  
Flux Limitation

AbstractThe flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ n t + u · ∇ n = Δ n - ∇ · ( n f ( | ∇ c | 2 ) ∇ c ) , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that $$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$ | f ( ξ ) | ≤ K f · ( ξ + 1 ) - α 2 holds for all $$\xi \ge 0$$ ξ ≥ 0 with some $$K_f>0$$ K f > 0 and $$\alpha >0$$ α > 0 , for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in $$L^\infty (\Omega \times (0,\infty ))$$ L ∞ ( Ω × ( 0 , ∞ ) ) with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of ($$\star $$ ⋆ ), this confirms that with regard to the possible emergence of blow-up phenomena, the choice $$f\equiv const.$$ f ≡ c o n s t . retains some criticality also in the presence of fluid interaction.

scholarly journals ON SINGULAR SOLUTIONS OF THE STATIONARY NAVIER-STOKES SYSTEM IN POWER CUSP DOMAINS

2021 ◽  
Vol 26 (4) ◽  
pp. 651-668
Author(s):  
Konstantinas Pileckas ◽  
Alicija Raciene
Keyword(s):  
Asymptotic Expansion ◽  
Stokes System ◽  
Boundary Value ◽  
Singular Solutions ◽  
Navier Stokes ◽  
Asymptotic Decomposition ◽  
Dirichlet Integral ◽  
Existence Of A Solution ◽  
Source Sink ◽  
Formal Asymptotic Expansion

The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.

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