Existence Of Self-similar Solutions Research Articles

Self-similar cosmological solutions in symmetric teleparallel theory: Friedmann-Lemaître-Robertson-Walker spacetimes

The existence of self-similar solutions is discussed in symmetric teleparallel $f(Q)$ theory for a Friedmann-Lema\^{\i}tre-Robertson-Walker background geometry with zero and nonzero spatial curvature. For the four distinct families of connections that describe the specific cosmology in symmetric teleparallel gravity, the functional form of $f(Q)$ is reconstructed. Finally, to see if the analogy with GR holds, we discuss the relation of the self-similar solutions with the asymptotic behavior of more general $f(Q)$ functions.

On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain

This paper is devoted to the mathematical analysis of a spatially homogeneous thermostatted kinetic theory framework with an unbounded activity domain. The framework consists of a partial integro-differential equation with quadratic nonlinearity where the domain of the activity variable is the whole real line. Specifically the mathematical analysis refers firstly to the existence and uniqueness of the solution for the related initial boundary value problem; Secondly the investigations are addressed to the existence of a class of self-similar solutions by employing the Fourier transform method. In particular the main result is obtained for a nonconstant interaction rate and a nonconstant force field. Conclusions and perspectives are discussed in the last section of the paper.

Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization

In this paper, we consider the egg-sperm chemotaxis model of coral with the incompressible fluid equations in the whole space. The existence of global mild solutions in scaling invariant spaces is proved with sufficient small initial data. Here the main tool we use is the implicit function theorem. Furthermore, we obtain the asymptotic stability of solutions when the time goes to infinity. Since the initial data could be in the weak L^{p}-spaces, we finally get the existence of self-similar solutions when the initial data are small homogeneous functions.

Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

We study localization occurring during high-speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition between Hadamard instability (caused by softening response) and the stabilizing effects of strain rate hardening. We consider a hyperbolic–parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition. The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is in four dimensions but has a fast–slow structure with respect to a small parameter capturing the strength of strain rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space.

Existence and uniqueness of mild solutions to the magneto-hydro-dynamic equations

In this paper, we consider the incompressible magneto-hydro-dynamic equations in the whole space. We first show that there exist global mild solutions with small initial data in the scaling invariant space. The main technique we have used is implicit function theorem which yields necessarily continuous dependence of solutions for the initial data. Moreover, we gain the asymptotic stability of solutions as the time goes to infinity. Finally, as a byproduct of our construction of solutions in the weak L p -spaces, the existence of self-similar solutions was established provided the initial data are small homogeneous functions.

Lie symmetry analysis of electron–electromagnetic wave interaction under condition of the anomalous Doppler effect

Lie symmetry analysis is applied for a problem of interaction of electron cyclotron oscillators with a slow electromagnetic wave under condition of the anomalous Doppler effect. This analysis reveals scaling invariance of the system and existence of self-similar solutions which describe amplification of a short electromagnetic pulse with its subsequent compression. The results of theoretical analysis are confirmed by numerical simulations.

Erratum to: Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels

We apologize to the readers for the following serious misprint in the above article that appears in the abstract and in the statement of the main result. Our result on the existence of self-similar solutions with fat tails is not valid for all γ ∈ (−∞, 1) but only for γ ∈ [0, 1). This is also in accordance with the title, because obviously only for the latter values of γ the kernel is locally bounded. We would also like to take this opportunity to correct a few minor mistakes and typos in the remainder of the paper. 1. In the proof of Lemma 2.2 the estimate of Q [H ] (X, t) in the norm ‖·‖Xρ goes as follows: Due to (9) and (12) we find ∫ R

Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels

The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (x γ + y γ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem.

The Reversing of Interfaces in Slow Diffusion Processes with Strong Absorption

This paper considers a family of one-dimensional nonlinear diffusion equations with absorption. In particular, the solutions that have interfaces that change their direction of propagation are examined. Although this phenomenon of reversing interfaces has been seen numerically, and some special exact solutions have been obtained, there was previously no analytical insight into how this occurs in the general case. The approach taken here is to seek self-similar solutions local to the interface and local to the reversing time. The analysis is split into two parts, one for the solution prior to the reversing time and the other for the solution after the reversing time. In each case the governing PDE is reduced to an ODE by introducing a self-similar coordinate system. These ODEs do not readily admit any nontrivial exact solutions and so the asymptotic behavior of solutions is studied. By doing this the adjustable parameters, or degrees of freedom, which may be used in a numerical shooting scheme are determined. A numerical algorithm is then proposed to furnish solutions to the ODEs and hence the PDE in the limit of interest. As examples of physical problems in which a PDE of this type may be used as a model the authors study the spreading of a viscous film under gravity and subject to evaporation, the dispersion of a population, and a nonlinear heat conduction problem. The numerical algorithm is demonstrated using these examples. Results are also given on the possible existence of self-similar solutions and types of reversing behavior that can be exhibited by PDEs in the family of interest.

Infinite energy solutions of the surface quasi-geostrophic equation

We study the formation of singularities of a 1D non-linear and non-local equation. We show that this equation provides solutions of the surface quasi-geostrophic equation with infinite energy. The existence of self-similar solutions and the blow-up for classical solutions are shown.

Self-similarity and uniqueness of solutions for semilinear reaction-diffusion systems

We study the well posedness of the initial-value problem for a coupled semilinear reaction-diffusion system in Marcinkiewicz spaces $L^{(p_{1}, \infty)}(\Omega)\times L^{(p_{2}, \infty)}(\Omega)$. The exponents $p_{1},p_{2}$ of the initial-value space are chosen to allow the existence of self-similar solutions (when $\Omega=\mathbb{R}^{n}$). As a nontrivial consequence of our coupling-term estimates, we prove the uniqueness of solutions in the scaling invariant class $C([0,\infty);L^{p_{1}}(\Omega)\times L^{p_{2}}(\Omega))$ regardless of their size and sign. We also analyze the asymptotic stability of the solutions, show the existence of a basin of attraction for each self-similar solution and that solutions in $L^{p_{1} }\times L^{p_{2}}$ present a simple long-time behavior.

On the Self-Similar Asymptotics for Generalized Nonlinear Kinetic Maxwell Models

Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economics, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. A new application of multi-linear models to economics and social dynamics is discussed.

Existence of solutions to the convection problem in a pseudomeasure-type space

We study the well-posedness of a convection problem in a pseudomeasure-type space PM a , without assuming that the gravitational field is bounded. Considering the PM a space with right homogeneity, the existence of self-similar solutions is proved. Finally, an analysis about asymptotic stability is made.

K-defects as compactons

We argue that topological compactons (solitons with compact support) may be quite common objects if k-fields, i.e., fields with nonstandard kinetic term, are considered, by showing that even for models with well-behaved potentials the unusual kinetic part may lead to a power-like approach to the vacuum, which is a typical signal for the existence of compactons. The related approximate scaling symmetry as well as the existence of self-similar solutions are also discussed. As an example, we discuss domain walls in a potential Skyrme model with an additional quartic term, which is just the standard quadratic term to the power two. We show that in the critical case, when the quadratic term is neglected, we get the so-called quartic ϕ4 model, and the corresponding topological defect becomes a compacton. Similarly, the quartic sine-Gordon compacton is also derived. Finally, we establish the existence of topological half-compactons and study their properties.

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong space is the null solution, infinitely many self-similar solutions do exist in weak- spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.

Self-Similar Solutions To The Oort–Hulst–Safronov Coagulation Equation

The existence of self-similar solutions with a finite first moment is established for the Oort–Hulst–Safronov coagulation equation when the coagulation kernel is given by $a(y,y_*)=y^\lambda+y_*^\lambda$ for some $\lambda\in (0,1)$. The corresponding self-similar profiles are compactly supported and have a discontinuity at the edge of their support.

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established.

Well-posedness and asymptotic behaviour for the convection problem in

We analyse the well-posedness of the initial value problem for a convection problem. Mild solutions are obtained in the weak- spaces and the existence of self-similar solutions is shown, while the only small self-similar solution in the Lebesgue space is the null solution. The asymptotic stability of solutions is analysed and, as a consequence, a criterion of self-similarity persistence at large times is obtained.

Nonexistence of self-similar solutions containing a black hole in a universe with a stiff fluid or scalar field or quintessence

We consider the possible existence of self-similar solutions containing black holes in a Friedmann background with a stiff fluid or a scalar field. We carefully study the relationship between the self-similar equations in these two cases and emphasize the crucial role of the similarity horizon. We show that there is no self-similar black hole solution surrounded by an exact or asymptotically flat Friedmann background containing a massless scalar field. This result also applies for a scalar field with a potential, providing the universe is decelerating. However, if there is a potential and the universe is accelerating (as in the quintessence scenario), the result only applies for an exact Friedmann background. This extends the result previously found in the stiff fluid case and strongly suggests that accretion onto primordial black holes is ineffective even during scalar field domination. It also contradicts recent claims that such black holes can grow appreciably by accreting quintessence. Appreciable growth might be possible with very special matter fields but this requires ad hoc and probably unphysical conditions.

Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations

We analyze the well-posedness of the initial-value problem for the semilinear equation in Marcinkiewicz spaces $L^{(p,\infty)}$. Mild solutions are obtained in spaces with the right homogeneity to allow the existence of self-similar solutions. As a consequence of our results we prove that the class $C([0,T);L^{p}(\Omega)),\ 0 < T\leq\infty, \ p={\frac{n(\rho-1)}{2\gamma }},$ $\Omega=R^{n},$ has uniqueness of solutions (including large solutions) obtained in [19], [20] and [8]. The asymptotic stability of solutions is obtained, and as a consequence, a criterion for self-similarity persistence at large times is obtained.