Existence and stability of steady noncharacteristic solutions on a
finite interval of full compressible **Navier**-**Stokes** **equations**

**equation**of state possessing a convex entropy for which there holds nonuniqueness of solutions. Expand abstract.

**equation**of state possessing a convex entropy for which there holds nonuniqueness of solutions. This is associated with instability and Hopf bifurcation to time-periodic solutions.

10/10 relevant

arXiv

A geometric look at momentum flux and stress in fluid mechanics

**equations**of motion for a variety of inviscid fluid models -- compressible and incompressible Euler equations, Lagrangian-averaged Euler-$\alpha$ equations, magnetohydrodynamics and shallow-water models -- using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the

**Navier**--

**Stokes**

**equations**for viscous fluids and of the Oldroyd-B model for visco-elastic fluids.

4/10 relevant

arXiv

Low Order Finite Element Methods for the **Navier**-Stokes-Cahn-Hilliard
**Equations**

**Navier**-Stokes-Cahn-Hilliard

**equations**, which have been established as a promising phase field modelling approach for simulation of immiscible multiphase flows. Expand abstract.

**Navier**-Stokes-Cahn-Hilliard equations, which have been established as a promising phase field modelling approach for simulation of immiscible multiphase flows. The present study suggests that traditional

**Navier**-Stokes-Cahn-Hilliard models do not allow for surface tension effects to be neglected due to the presence of the surface tension parameter in the Cahn-Hilliard

**equation**. This motivates the proposed formulation, which allows surface tension effects to be changed without affecting the behaviour of the phase transport. Two methods are proposed: The first uses stabilised SUPG/PSPG linear elements, while the second is based on mixed Taylor-Hood elements. The proposed models are applied to a number of benchmark and example problems, including both capillary regime in which surface tension effects are dominant, and inertial regime in which surface tension effects are negligibly small. All results obtained agree very well with reference solutions.

10/10 relevant

arXiv

Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems

**Navier**-

**Stokes**

**equations**, known as Oseen

**equation**s, is presented. Expand abstract.

**Navier**-

**Stokes**equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order k+1 is obtained for velocity, pressure and strain-rate tensor and a local postprocessing strategy is devised to construct an approximation of the velocity superconverging with order k+2, even for low-order polynomial approximations. A tutorial for the numerical solution of incompressible flow problems using HDG is presented, with special emphasis on the technical details required for its implementation.

7/10 relevant

arXiv

Second order pressure estimates for the Crank-Nicolson discretization of
the incompressible **Navier**-**Stokes** **Equations**

**Navier**-

**Stokes**

**equations**. Expand abstract.

**Navier**-

**Stokes**

**equations**. Second order estimates for the velocity error are long known, we prove that the pressure error is of the same order if considered at interval midpoints, confirming previous numerical evidence. For simplicity we first give a proof under high regularity assumptions that include nonlocal compatibility conditions for the initial data, then use smoothing techniques for a proof under reduced assumptions based on standard local conditions only.

10/10 relevant

arXiv

On self-similar solutions to degenerate compressible **Navier**-Stokes
**equations**

**Navier**-

**Stokes**

**equations**with degenerate density-dependent viscosity. Expand abstract.

**Navier**-

**Stokes**

**equations**with degenerate density-dependent viscosity. We prove both existence of expanders and non-existence of small shrinkers.

10/10 relevant

arXiv

Asymptotic stability of homogeneous solutions of incompressible
stationary **Navier**-**Stokes** **equations**

**Navier**-

**Stokes**

**equations**in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.

10/10 relevant

arXiv

Global regularity for the hyperdissipative **Navier**-**Stokes** **equation** below
the critical order

**Navier**-

**Stokes**

**equation**with fractional dissipation of order $\alpha\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^{\delta}} \leq M$, where $M$ is arbitrarily large and $\delta$ is arbitrarily small, there exists an explicit $\epsilon=\epsilon(M, \delta)>0$ such that the

**Navier**-

**Stokes**

**equations**with fractional order $\alpha$ has a unique smooth solution for $\alpha \in (\frac{5}{4}-\epsilon, \frac{5}{4}]$. This is related to a new stability result on smooth solutions of the

**Navier**-

**Stokes**

**equations**with fractional dissipation showing that the set of initial data and fractional orders giving rise to smooth solutions is open in $H^{5/4} \times (\frac 34, \frac{5}{4}]$.

10/10 relevant

arXiv

On the Serrin-type condition on one velocity component for the
**Navier**-**Stokes** **equations**

**Navier**-

**Stokes**

**equations**in $ \R^{3} $. We show that the Serrin-type condition imposed on one component of the velocity $ u_3\in L^p(0,T; L^q(\R^{3} ))$ for with $ \frac{2}{p}+ \frac{3}{q}

10/10 relevant

arXiv

Existence of steady solutions of Newtonian and Non-Newtonian fluids with right hand sides beyond duality

**Navier**-

**Stokes**

**equations**as well as non-Newtonian fluids of power law type with right-hand sides that are not in the natural existence class. Expand abstract.

**Navier**-

**Stokes**

**equations**as well as non-Newtonian fluids of power law type with right-hand sides that are not in the natural existence class. The estimates for the

**Navier**-

**Stokes**

**equation**follows by a newly developed comparison method. For the estimates we make use of a newly developed solenoidal Lipschitz truncation (more explicit a relative truncation) that preserves zero boundary values. The estimates imply respective existence results for very weak right hand sides.

10/10 relevant

arXiv