It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E={ℝ}^{n+1}$ for which $\partial E={ℝ}^n\times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ${ℝ}^n\times [0,\infty )$.

Elastica and inextensible flows of curves play an important role in practical applications. In this paper, we construct a new characterization of inextensible flows by using elastica in space. The inextensible flow is completely determined for any space-like curve in de Sitter space [...] S 1 3 ${𝕊}_1^3$. Finally, we give some characterizations for curvatures of a space-like curve in de Sitter space [...] S 1 3 ${𝕊}_1^3$.

A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$, $n+1/2$ and $n+1$. The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level...

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T4$ configurations. We then use this to construct weak solutions to the unstable interface...

We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target manifold $N$, $$\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\frac{\partial }{\partial t}\phi ={\tau }_g\phi ,$$ where $\alpha$ is a (possibly time-dependent) positive coupling constant. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of $\phi$ a-priori by choosing $\alpha$ large enough. Moreover, it suffices to bound the curvature...

Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^q$-estimates of second order derivatives uniformly in the angular and translational velocities,...

We construct some nondecreasing quantities associated to the first eigenvalue of $-{\Delta }_{\phi }+cR$ $(c\ge \frac{1}{2}(n-2)/(n-1))$ along the Yamabe flow, where ${\Delta }_{\phi }$ is the Witten-Laplacian operator with a $C^2$ function $\phi$. We also prove a monotonic result on the first eigenvalue of $-{\Delta }_{\phi }+\frac{1}{4}(n/(n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-{\Delta }_{\phi }+c{R}^a$ with $a\in (0,1)$ along the Yamabe flow.

This study deals with a numerical solution of a 2D flows of a compressible viscous fluids in a convergent channel for low inlet airflow velocity. Three governing systems – Full system, Adiabatic system, Iso-energetic system$basedontheNavier-Stokesequationsforlaminarflowaretested.Thenumericalsolutionisrealizedbyfinitevolumemethodandthepredictor-correctorMacCormackschemewithJamesonartificialviscosityusingagridofquadrilateralcells.TheunsteadygridofquadrilateralcellsisconsideredintheformofconservationlawsusingArbitraryLagrangian-Eulerianmethod.Thenumericalresults,acquiredfromadevelopedprogram,arepresentedforinletvelocity$u=4.12 ms-1$andReynoldsnumberRe=$4 103$.$

Kočergin introduced in 1975 a class of smooth flows on the two torus that are mixing. When these flows have one fixed point, they can be viewed as special flows over an irrational rotation of the circle, with a ceiling function having a power-like singularity. Under a Diophantine condition on the rotation’s angle, we prove that the special flows actually have a $t^{-\eta }$-speed of mixing, for some $\eta \>0$.

Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_p^{2,1}\left(\Omega \times (0,T)\right)$ and the pressure q satisfies $\nabla q\in L_p\left(\Omega \times (0,T)\right)$ for p ≥ 7/3.

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness...