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Register a new userA weighted setting for the stationary Navier Stokes equations under singular forcing
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In two dimensions, we show existence of solutions to the stationary Navier Stokes equations on weighted spaces $mathbf{H}^1_0(omega,Omega) times L^2(omega,Omega)$, where the weight belongs to the Muckenhoupt class $A_2$. We show how this theory can be applied to obtain a priori error estimates for approximations of the solution to the Navier Stokes problem with singular sources.
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In Lipschitz two and three dimensional domains, we study the existence for the so--called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to $H^{-1}(varpi,Omega)$, where $varpi$ is a weight in the Muckenhoupt class $A_2$ that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and $varpi^{-1} in A_1$, show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability and local efficiency.
In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier Stokes equations with singular sources on Lipschitz, but not necessarily convex, polygonal domains. Under a smallness assumption on the continuous and discrete solutions, we prove that the devised error estimator is reliable and locally efficient. We illustrate the theory with numerical examples.
This work studies the system of $3D$ stationary Navier-Stokes equations. Several Liouville type theorems are established for solutions in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions of the stationary Navier-Stokes equations are identically zero. This result covers the important case that solutions may decay to zero with different rates in different spatial directions, and some these rates could be significantly slow. In the un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions, this work also provides several other important Liouville type theorems for solutions in weighted mixed-norm Lebesgue spaces. To prove the results, we establish some new results on mixed-norm and weighted mixed-norm estimates for Navier-Stokes equations. All of these results are new and could be useful in other studies.
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy develop nearly singular solutions at the origin. This nearly singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in cite{Hou-euler-2021}. One important feature of the potential Euler singularity is that the solution develops nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Navier-Stokes equations develop nearly singular scaling properties with maximum vorticity increased by a factor of $10^7$. Moreover, the nearly self-similar profiles seem to be very stable to the small perturbation of the initial data. However, the 3D Navier-Stokes equations with our initial data do not develop a finite time singularity due to the development of a mild two-scale structure in the late stage, which eventually leads to viscous dominance over vortex stretching. To maintain the balance between the vortex stretching term and the diffusion term, we solve the 3D Navier-Stokes equations with a time-dependent viscosity roughly of order $O(|log(T-t)|^{-3})$ in the late stage. We present strong numerical evidence that the 3D Navier-Stokes equations with such time-dependent viscosity develop a finite time singularity.
In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $epsilon$, which induces a small divergence and the time step $delta$t tend to zero with a proportionality constraint $epsilon$ = $lambda$$delta$t. Finally, when $eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{e}sum{e} Dans ce travail nous analysons un sch{e}ma de projection vectorielle (voir [1]) pour traiter le d{e}placement dun corps solide dans un fluide visqueux incompressible dans le cas o` u linteraction du fluide sur le solide est n{e}gligeable. La pr{e}sence de lobstacle dans le domaine solide est mod{e}lis{e}e par une m{e}thode de p{e}nalisation. Nous montrons la stabilit{e} du sch{e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $epsilon$ qui assure une faible divergence et le pas de temps $delta$t tendent vers 0 avec une contrainte de proportionalit{e} $epsilon$ = $lambda$$delta$t. Finalement nous montrons que leprob{`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{`i} ere immerg{e}e quand le param etre de p{e}nalisation $eta$ tend vers 0.
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