Searcharxiv⌕ Search

arXiv subjects

Peter Wittwer

Publications and source records attributed to Peter Wittwer.

At least 19 recordsLinked to original sources
arXiv2022

A new instability in clustering dark energy?

In this paper, we study the effective field theory (EFT) of dark energy for the $k$-essence model beyond linear order. Using particle-mesh $N$-body simulations that consistently solve the dark energy evolution on a grid, we find that the next-to-leading order in the EFT expansion, which comprises the terms of the equations of motion that are quadratic in the field variables, gives rise to a new instability in the regime of low speed of sound (high Mach number). We rule out the possibility of a numerical artefact by considering simplified cases in spherically and plane symmetric situations analytically. If the speed of sound vanishes exactly, the non-linear instability makes the evolution singular in finite time, signalling a breakdown of the EFT framework. The case of finite (but small) speed of sound is subtle, and the local singularity could be replaced by some other type of behaviour with strong non-linearities. While an ultraviolet completion may cure the problem in principle, there is no reason why this should be the case in general. As a result, for a large range of the effective speed of sound $c_s$, a linear treatment is not adequate.

astro-ph.CO↗
arXiv2021

A new nonlinear instability for scalar fields

In this letter we introduce the non-linear partial differential equation (PDE) $\partial^2_{\tau} \pi \propto (\vec\nabla \pi)^2$ showing a new type of instability. Such equations appear in the effective field theory (EFT) of dark energy for the $k$-essence model as well as in many other theories based on the EFT formalism. We demonstrate the occurrence of instability in the cosmological context using a relativistic $N$-body code, and we study it mathematically in 3+1 dimensions within spherical symmetry. We show that this term dominates for the low speed of sound limit where some important linear terms are suppressed.

astro-ph.CO↗
arXiv2018

Nonlinear stability at the Eckhaus boundary

The real Ginzburg-Landau equation possesses a family of spatially periodic equilibria. If the wave number of an equilibrium is strictly below the so called Eckhaus boundary the equilibrium is known to be spectrally and diffusively stable, i.e., stable w.r.t. small spatially localized perturbations. If the wave number is above the Eckhaus boundary the equilibrium is unstable. Exactly at the boundary spectral stability holds. The purpose of the present paper is to establish the diffusive stability of these equilibria. The limit profile is determined by a nonlinear equation since a nonlinear term turns out to be marginal w.r.t. the linearized dynamics.

math.AP↗
arXiv2017

Existence and uniqueness of steady weak solutions to the Navier-Stokes equations in $\mathbb{R}^2$

The existence of weak solutions to the stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ is proven. This particular geometry was the only case left open since the work of Leray in 1933. The reason is that due to the absence of boundaries the local behavior of the solutions cannot be controlled by the enstrophy in two dimensions. We overcome this difficulty by constructing approximate weak solutions having a prescribed mean velocity on some given bounded set. As a corollary, we obtain infinitely many weak solutions in $\mathbb{R}^2$ parameterized by this mean velocity, which is reminiscent of the expected convergence of the velocity field at large distances to any prescribed constant vector field. This explicit parameterization of the weak solutions allows us to prove a weak-strong uniqueness theorem for small data. The question of the asymptotic behavior of the weak solutions remains however open, when the uniqueness theorem doesn't apply.

math.AP↗
arXiv2015

Optimal asymptotic behavior of the vorticity of a viscous flow past a two-dimensional body

The asymptotic behavior of the vorticity for the steady incompressible Navier-Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero. It is well known that the asymptotic behavior of the velocity field is given by the fundamental solution of the Oseen system which is the linearization of the Navier-Stokes equation around $\boldsymbol{u}_{\infty}$. The vorticity has the property of decaying algebraically inside a parabolic region called the wake and exponentially outside. The previously proven asymptotic expansions of the vorticity are relevant only inside the wake because everywhere else the remainder is larger than the asymptotic term. Here we present an asymptotic expansion that removes this weakness. Surprisingly, the found asymptotic term is not given by the Oseen linearization and has a power of decay that depends on the data. This strange behavior is specific to the two dimensional problem and is not present in three dimensions.

math.AP↗
arXiv2015

On the stationary Navier-Stokes equations in the half-plane

We consider the stationary incompressible Navier-Stokes equation in the half-plane with inhomogeneous boundary condition. We prove existence of strong solutions for boundary data close to any Jeffery-Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery-Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove existence of weak solutions, as well as weak-strong uniqueness for small data.

math.AP↗
arXiv2014

Generalized scale-invariant solutions to the two-dimensional stationary Navier-Stokes equations

New explicit solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^{2}\setminus\left\{ \boldsymbol{0}\right\}$ are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant under a particular combination of the scaling and rotational symmetries. They are the only solutions invariant under this new symmetry in the same way as the Hamel solutions are the only scale-invariant solutions. While the Hamel solutions are parameterized by a discrete parameter $n$, the flux $Φ$ and an angle $θ_{0}$, the new solutions generalize the Hamel solutions by introducing an additional parameter $a$ which produces a rotation. The new solutions decay like $\left|\boldsymbol{x}\right|^{-1}$ as the Hamel solutions, and exhibit spiral behavior. The new variety of asymptotes induced by the existence of these solutions further emphasizes the difficulties faced when trying to establish the asymptotic behavior of the Navier-Stokes equations in a two-dimensional exterior domain or in the whole plane.

math.AP↗
arXiv2013

Asymptotic behaviour of solutions to the stationary Navier-Stokes equations in two dimensional exterior domains with zero velocity at infinity

We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier-Stokes equations in a two-dimensional exterior domain. More precisely, we find the asymptotic behaviour for such solutions in the case where the net force on the boundary of the domain is non-zero. In contrast to the three dimensional case, where the asymptotic behaviour is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width $|\boldsymbol{x}|^{2/3}$, the velocity decays like $|\boldsymbol{x}|^{-1/3}$, whereas outside the wake, it decays like $|\boldsymbol{x}|^{-2/3}$. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the non-zero and the zero net force case.

physics.flu-dyn↗
arXiv2012

Artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane

We discuss artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane, for a range of low Reynolds numbers. When truncating the half-plane to a finite domain for numerical purposes, artificial boundaries appear. We present an explicit Dirichlet condition for the velocity at these boundaries in terms of an asymptotic expansion for the solution to the problem. We show a substantial increase in accuracy of the computed values for drag and lift when compared with results for traditional boundary conditions. We also analyze the qualitative behavior of the solutions in terms of the streamlines of the flow. The new boundary conditions are universal in the sense that they depend on a given body only through one constant, which can be determined in a feed-back loop as part of the solution process.

physics.flu-dyn↗
arXiv2012

Asymptotics of solutions for a basic case of fluid structure interaction

We consider the Navier--Stokes equations in a half-plane with a drift term parallel to the boundary and a small source term of compact support. We provide detailed information on the behavior of the velocity and the vorticity at infinity in terms of an asymptotic expansion at large distances from the boundary. The expansion is universal in the sense that it only depends on the source term through some multiplicative constants. This expansion is identical to the one for the problem of an exterior flow around a small body moving at constant velocity parallel to the boundary, and can be used as an artificial boundary condition on the edges of truncated domains for numerical simulations.

math-ph↗
arXiv2011

Decay estimates for steady solutions of the Navier-Stokes equations in two dimensions in the presence of a wall

Let w be the vorticity of a stationary solution of the two-dimensional Navier-Stokes equations with a drift term parallel to the boundary in the half-plane -\infty 1, with zero Dirichlet boundary conditions at y=1 and at infinity, and with a small force term of compact support. Then, |xyw(x,y)| is uniformly bounded in the half-plane. The proof is given in a specially adapted functional framework and complements previous work.

math-ph↗
arXiv2011

Asymptotic description of solutions of the exterior Navier Stokes problem in a half space

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall.

math.AP↗
arXiv2009

Graviton production in anti-de Sitter braneworld cosmology: A fully consistent treatment of the boundary condition

In recent work by two of us, [Durrer & Ruser, PRL 99, 071601 (2007); Ruser & Durrer PRD 76, 104014 (2007)], graviton production due to a moving spacetime boundary (braneworld) in a five dimensional bulk has been considered. In the same way as the presence of a conducting plate modifies the electromagnetic vacuum, the presence of a brane modifies the graviton vacuum. As the brane moves, the time dependence of the vacuum leads to particle creation via the so called 'dynamical Casimir effect'. In our previous work a term in the boundary condition which is linear in the brane velocity has been neglected. In this work we develop a new approach which overcomes this approximation. We show that the previous results are not modified if the brane velocity is low.

astro-ph.CO↗
arXiv2009

Braneworlds, graviton production, dynamical Casimir effect

If our Universe is a 3+1 brane in a warped 4+1 dimensional bulk so that its expansion can be understood as the motion of the brane in the bulk, the time dependence of the boundary conditions for arbitrary bulk fields can lead to particle creation via the dynamical Casimir effect. In this talk I report results for the simplest such scenario, when the only particle in the bulk is the graviton and the bulk is the 5 dimensional anti-de Sitter spacetime.

astro-ph.CO↗
arXiv2008

Adaptive boundary conditions for exterior stationary flows in three dimensions

Recently there has been an increasing interest for a better understanding of ultra low Reynolds number flows. In this context we present a new setup which allows to efficiently solve the stationary incompressible Navier-Stokes equations in an exterior domain in three dimensions numerically. The main point is that the necessity to truncate for numerical purposes the exterior domain to a finite sub-domain leads to the problem of finding so called "artificial boundary conditions" to replace the conditions at infinity. To solve this problem we provide a vector filed that describes the leading asymptotic behavior of the solution at large distances. This vector field depends explicitly on drag and lift which are determined in a self-consistent way as part of the solution process. When compared with other numerical schemes the size of the computational domain that is needed to obtain the hydrodynamic forces with a given precision is drastically reduced, which in turn leads to an overall gain in computational efficiency of typically several orders of magnitude.

physics.flu-dyn↗
arXiv2003

Some exact results for Boltzmann's annihilation dynamics

The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Exact analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with non-universal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in presence of static traps. The analytical predictions are shown to be in excellent agreement with the results of two-dimensional molecular dynamics simulations.

cond-mat.stat-mech↗
arXiv1999

Asymptotics of solutions in nA+nB->C reaction Diffusion systems

We analyze the long time behavior of initial value problems that model a process where particles of type A and B diffuse in some substratum and react according to $nA+nB\to C$. The case n=1 has been studied before; it presents nontrivial behavior on the reactive scale only. In this paper we discuss in detail the cases $n>3$, and prove that they show nontrivial behavior on the reactive and the diffusive length scale.

math-ph↗