The aerodynamics of the blast wave produced by laser ablation is studied using the piston analogy. The unsteady one-dimensional gasdynamic equations governing the flow are solved under assumption of self-similarity. The solutions are utilized to obtain analytical expressions for the velocity, density, pressure and temperature distributions. The results predict all the experimentally observed fe...

We study in this article the equivariant Schrödinger map equation in dimension 2, from the Euclidean plane to the sphere. A family of self-similar solutions is constructed; this provides an example of regularity breakdown for the Schrödinger map. These solutions do not have finite energy, and hence do not fit into the usual framework for solutions. For data of infinite energy but small in some ...

In this paper, we consider Barenblatt’s k– model for turbulence. For the case of equal diffusion coefficients α and β, Barenblatt found explicit compactly supported self-similar solutions. From these, we obtain compactly supported solutions for α 6= β by transforming the equations into a four-dimensional quadratic system and verifying a transversality condition for a saddle-point connection. Th...

We prove the existence of self-similar solutions to the Fradkov model for twodimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbours (topological class). For the proof we introduce a finite maximal topological class and study an appropriate upwind-discretization of the ti...

We explore semi-complete self-similar solutions for the polytropic gas dynamics involving self-gravity under spherical symmetry, examine behaviours of the sonic critical curve, and present new asymptotic collapse solutions that describe ‘quasi-static’ asymptotic behaviours at small radii and large times. These new ‘quasi-static’ solutions with divergent mass density approaching the core can hav...

Abstract. We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function u has eigenvalues strictly uniformly between −1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one-to-one correspondence to functions of homogenous of degree 2 with th...

This paper discusses the diffusion equation with a damping term as follows ut = div(|Dum|p−2Dum)− u1 |Du|1 , where p > 2,m > 1, and p > p1, q1+p1m > m(p−1) > 1. By the standard Picard iteration method, a sufficient condition is given to the existence of the singular self-similar solutions. Moreover, the paper gives a classification for these singular self-similar solutions. Key–Words: Diffusion...

We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇ · (u∇K ∗ u), where the kernel K(x) = |x|. The dynamics of the blowup exhibits self-similar behavior in which zero mass concentrates at the core at the blowup time. Computations are performed in Rn for n between 2 and 10 using a method based on characteristics. In all cases studied, the ...

Existence of self-similar solutions to the Oort-Hulst-Safronov coagulation equation with multiplicative coagulation kernel is established. These solutions are given by s(t)−τ ψτ (y/s(t)) for (t, y) ∈ (0, T )×(0,∞), where T is some arbitrary positive real number, s(t) = ((3−τ)(T − t))−1/(3−τ) and the parameter τ ranges in a given interval [τc, 3). In addition, the second moment of these self-sim...