Self-similar Blow Research Articles

Backward self-similar solutions for compressible Navier–Stokes equations

This article is devoted to backward self-similar blow up solutions of the compressible Navier–Stokes equations with radial symmetry. We show that such solutions cannot exist if they either have finite energy, or satisfy appropriate smallness conditions. Furthermore, numerical simulations lead us to the conjecture that such solutions do not exist.

Dynamical behavior near explicit self-similar blow–up solutions for the Born–Infeld equation

This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one-dimensional Born–Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow-up solutions; meanwhile, Lyapunov nonlinear stability of those self-similar blow-up solutions is given inside a strictly proper subset of the backward light cone.

On the energy behavior of locally self-similar blowup for the Euler equation

In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of possible blowup.

On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f 0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f 0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f 0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than $${\frac{1}{2}}$$ . The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

Self-similar blow up with a continuous range of values of the aggregated mass for a degenerate Keller-Segel system

In this paper we show that there exist radial solutions of the Keller-Segel system with porous medium like diffusion and critical exponents that blow up in a self-similar manner with a continuous range of masses. The situation is very different from the one that takes place for the usual Keller-Segel system with semilinear diffusion, that for critical nonlinearities yields blow up with a discrete set of values for the mass.

Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system

The three dimensional gravitational Vlasov-Poisson system ∂ t f + v ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+v\cdot \nabla _x f-E_f\cdot \nabla _vf=0 , where E f ( x ) = ∇ x ϕ f ( x ) E_f(x)=\nabla _x \phi _f(x) , Δ x ϕ f = ρ f ( x ) \Delta _x\phi _f=\rho _f(x) , ρ f ( x ) = ∫ R N f ( x , v ) d v \rho _f(x)=\int _{\mathbb {R}^N} f(x,v)dv , describes the mechanical state of a stellar system subject to its own gravity. Smooth initial data yield unique global in time solutions from a celebrated result by Pfaffelmoser. There exists a hierarchy of physical models which aim at taking into account further relativistic effects. The simplest one is the three dimensional relativistic gravitational Vlasov-Poisson system ∂ t f + v 1 + | v | 2 ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+\frac {v}{\sqrt {1+|v|^2}}\cdot \nabla _x f-E_f\cdot \nabla _vf=0 which we study here. A striking feature as observed by Glassey and Schaeffer is that this system now admits finite blow up solutions. Nevertheless, the existence argument is purely obstructive and provides no insight into the description of the singularity formation. We exhibit in this paper a family of finite time blow up self-similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations. Our analysis applies to the four dimensional gravitational Vlasov-Poisson system as well.

Vortex dynamics in perfect fluids

I review the current status of a problem, relevant to both plasma physics and ordinary fluid mechanics, namely the long-time behaviour of solutions of the perfect fluid equations. In two space dimensions, thanks in particular to the work of D. Montgomery, the situation is now quite clear, since one expects the formation at long times of large vortices in a background of potential flow. In three dimensions, the situation is blurred, although its understanding is a central issue for fully developped turbulence. I present some new estimates for a possible scenario of self-similar blow up of solutions of 3D Euler. That turns out to be a rather subtle question, if one tries to stay consistent with the conservation of circulation and of energy.

Singularities in complex interfaces

We analyse an equation describing the motion of the material interface between two fluids in a pressure field. The interface can be expressed as the image of the unit circle under a certain time depending conformal map. This conformal transformation maps the exterior of the unit circle onto the region occupied by one of the fluids. The conformal map has singularities in the unit disc. As long as these singularities are close to the origin, the complicated non-local equation governing the evolution of the conformal map can be approximated by a somewhat simpler, local equation. We prove that there exist self-similar solutions of this equation, that they have singularities away from the origin, that these singularities hit in finite time the unit circle and that the self-similar blow up is stable to perturbations that respect the symmetry of the self-similar profile.

Stable self-similar blow up for energy subcritical wave equations

We consider the semilinear wave equation partial derivative(2)(t)psi - Delta psi = vertical bar psi vertical bar(p-1)psi for 1 0 and kappa(p) is a p-dependent constant. We prove that the blow up described by psi(T) is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag. The method of proof is quite robust and can be applied to other self-similar blow up problems as well, even in the energy supercritical case.