Peter Wittwer

École Polytechnique Fédérale de Lausanne | EPFL · Mathematics Section

We consider the problem of a small 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 steady Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. In th...

In this paper, we establish the local, global existence and large-time behaviors of strong solutions to the free boundary problem of the planar magnetohydrodynamic equations with degenerate viscosity coefficient. Only the initial energy at the basic level is required to be small. The main difficulties are the degeneracy of the system near the free...

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 t...

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 consider the time-dependent Oseen sytem with rotational terms. This system is a linearized model for the ow of a viscous incompressible uid around a rigid body moving at a constant velocity and rotating with constant angular velocity. We present results on temporal and spatial decay of solutions to this system in the whole space. The spatial asy...

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...

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 co...

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 invarian...

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 contra...

We consider the problem of a body moving within an incompressible ‡uid 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 in…nity. Here we prove exist...

We establish some regularity criteria for the incompressible Navier-Stokes equations in a bounded three-dimensional domain concerning the quotients of the pressure, the velocity field and the pressure gradient.

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 asympt...

We consider the stationary incompressible Navier Stokes equation in the exterior of a disk B with non-zero Dirichlet boundary conditions on the disk and zero boundary conditions at infinity. We prove the existence of solutions for an open set of boundary conditions without symmetry.

We consider the problem of a body moving within an incompressible uid at constant speed parallel to a wall, in an otherwise unbounded domain. We give a detailed description of the asymptotic behavior on the uid ow in a half-space using as a starting point the theory of the existence of solutions which uses the coordinate perpendicularly to the wall...

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 sens...

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 ca...

Let omega 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 Omega(+) = {(x, y) is an element of R-2 vertical bar y > 1}, with zero Dirichlet boundary conditions at y = 1 and at infinity, and with a small force term of compact support. Then vertical b...

We construct solutions for the Navier-Stokes e quations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with th...

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. Here we prove exis...

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, w...

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 th...

Measurements using error terms to describe the imperfections of test equipment have been used for a long time, and there are many methods utilized to model the error terms. In all these methods, after having determined the error terms on the basis of calibration measurements, the s-parameters of the device-under-test (DUT) are computed from the mea...

We derive the equation for the vorticity of the incompressible Oseen problem in a half plane with homogeneous (no slip) boundary conditions. The resulting equation is a scalar Oseen equation with certain Dirichlet boundary conditions which are determined by the incompressibility condition and the boundary conditions of the original problem. We prov...

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 numer...

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve...

We discuss a new numerical scheme involving adaptive boundary condi-tions which allows to compute, at very low Reynolds numbers, drag and lift of airfoils with rough surfaces efficiently with great pre-cision. As an example we present the numeri-cal implementation for an airfoil consisting of a line segment. The solution of the Navier-Stokes equati...

We discuss uid structure interaction for exterior ows, for Reynolds numbers ranging from about one to several thousand. New applications demand a better quantitative understanding of the details of such ows, and this has stimulated a revival of interest in this topic. Astonishingly, in spite of the apparent simplicity of low Reynolds number ows, th...

We consider the question of the existence of stationary solutions for the Navier Stokes equations describing the flow of a incompressible fluid past a semi-infinite flat plate at zero incidence angle. By using ideas from the theory of dynamical systems we analyze the vorticity equation for this problem and show that a symmetry-breaking term fits na...

The simulation of a stationary fluid flow past an obstacle by means of a lattice Boltz- mann method is discussed. The problem of finding appropriate boundary conditions on the boundaries of the truncated numerical domain is addressed by a method re- cently discussed in the literature, based on a truncated expansion of the solution. The iterative pr...

We consider stationary solutions of the incompressible Navier–Stokes equations in three dimensions. We give a detailed description of the fluid flow in a half-space through the construction of an inertial manifold for the dynamical system that one obtains when using the coordinate along the flow as a time.

We consider stationary solutions of the incompressible Navier–Stokes equations in two dimensions. We give a detailed description of the fluid flow in a half-plane through the construction of an inertial manifold for the dynamical system that one obtains when using the coordinate along the flow as a time.

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. This corresponds to studying the stationary fluid flow past a body. The necessity to truncate for numerical purposes the infinite exterior domain to a finite domain leads to the problem of finding appropriate...

The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. 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...

This paper is a supplementary section to [1]. Weshow that without any additional hypothesis the main result in [1] (Theorem 1) can be considerably strengthened. Note: This paper can not be read independently of [1]. The numbering of equations, theorems and propositions as well as cross-references used here have to be understoodasifthispaper were an...

We consider stationary solutions of the incompressible Navier–Stokes equations in two dimensions. We give a detailed description of the fluid flow in a half-plane by using a mathematical setup within which the idea of a change of type from an elliptic to a parabolic partial differential equation can be made precise.

This paper is a supplementary section to [1]. We show that without any additional hypothesis the main result in [1] (Theorem 1) can be considerably strengthened. Note. This paper cannot be read independently of [1]. The numbering of equations, theorems and propositions as well as cross-references used here have to be understood as if this paper wer...

We consider stationary solutions of the incompressible Navier-Stokes equations in two dimensions.

. This paper is a summary of some recent work by the authors on the renormalization of Hamiltonian systems. Applications include the construction of invariant tori and related sequences of closed periodic orbits. We also discuss problems related to the breakup of invariant tori, and some numerical results. The renormalization group transformation....

. In this paper we extend the technique of computer{assisted proofs to xed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete pr...

. We describe a renormalization group transformation that is related to the breakup of golden invariant tori in Hamiltonian systems with two degrees of freedom. This transformation applies to a large class of Hamiltonians, is conceptually simple, and allows for accurate numerical computations. In a numerical implementation, we find a nontrivial fix...

. In this paper we present a computer#assisted proof of the existence of a solution for the Feigenbaum equation '#x# = 1 # '#'##x##: There exist by now various such proofs in the literature. Although the one presented here is new, the main purpose of this paper is not to provide yet another version, but to give an easy#to#read and self contained in...

. We prove that the Renormalization Group transformation for the Laplace transform of the d = 3 Dyson--Baker hierarchical model has a nontrivial entire analytic fixed point whose zeros all lie on the imaginary axis. Sharp upper and lower bounds on 80 of these zeros are used to verify the assumptions made in reference [1]. Our proof is computer--ass...

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 sho...

The authors analyze iterations of maps on an interval with an added noise term, in the neighbourhood of an intermittency threshold. They rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise.

We prove the existence of a nontrivial Renormalization Group (RG) fixed point for the Dyson-Baker hierarchical model ind=3 dimensions. The single spin distribution of the fixed point is shown to be entire analytic, and bounded by exp(−const×t 6) for large real values of the spint. Our proof is based on estimates for the zeros of a RG fixed point fo...

. We analyze the long time behavior of an initial value problem that models a chemical reaction--diffusion process A+B ! C: The problem has previously been studied by G'alfi and R'acz [1], who predicted the critical indices associated with the reaction by using a scaling ansatz motivated by numerical simulations. In this paper we point out some dif...

In this paper we describe invariant geometrical structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero),...

. We prove that the Renormalization Group transformation for the Laplace transform of the d = 3 Dyson--Baker hierarchical model has a nontrivial entire analytic fixed point whose zeros all lie on the imaginary axis. Sharp upper and lower bounds on 80 of these zeros are used to verify the assumptions made in reference [11]. Our proof is computer--as...

We analyze the long time behavior of an initial value problem that models a chemical reaction-diffusion process A + B --> C. The problem has previously been studied by Gálfi and Rácz, who predicted the critical indices associated with the reaction by using a scaling ansatz motivated by numerical simulations. In this paper we point out some difficul...

We give a new proof for the existence of a non-Gaussian hierarchical renormalization group fixed point, using what could be called a beta-function for this problem. We also discuss the asymptotic behavior of this fixed point, and the connection between the hierarchical models of Dyson and Gallavotti.

In this paper we give an outline of a computer assisted proof in which we use an extended version of the Wilson-Kadanoff renormalization group scheme to get rigorous bounds on a critical exponent that is universal for a class of one parameter families F µ of hierarchical lattice systems in d = 3 dimensions. The parameter µ will be referred to as te...

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map . This map acts on functions that are jointly analytic in a position variable (t) and in the parameter () that controls the period doubling phenomenon. A fixed point * for this map is found. The usual renormalization group doubling operatorN acts on this functio...

A rigorous method is developed to handle the “large field problems” in the Wilson-Kadanoff renormalization group approach to critical lattice systems of unbounded spins. We use this method to study in a hierarchical approximation the non-Gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theori...

Existence and hyperbolicity of fixed points for the mapN p :f(x) →λ−1f p (λx), withf p p-fold iteration and λ=f p (0) are given forp large. These fixed points come close to being quadratic functions, and our proof consists in controlling perturbation theory about quadratic functions.

We describe a computer-assisted proof of the existence of a fixed point φ of the doubling transformation (functional composition) for area-preserving maps of the plane: φ is obtained from the solution S to a fixed-point equation RS=S for generating functions of area-preserving maps. This establishes, with mathematical rigor, an important part of Fe...