Self-similar Solutions Research Articles

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

Spectral theory and self-similar blowup in wave equations

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account of the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.

Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.

Mathematical analysis and asymptotic predictions of chemical-driven swimming living organisms in weighted networks

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the L1- convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows.

Homothetic $$\alpha $$-Self-Similar Solutions to the Mean Curvature Flow in Minkowski 3-Space

Homothetic $$\alpha $$-Self-Similar Solutions to the Mean Curvature Flow in Minkowski 3-Space

Two-Layer Equilibrium Model of Miscible Inhomogeneous Fluid Flow

Two-layer flow of a density-stratified fluid with mass transfer between the layers is considered. In the Boussinesq approximation, the equations of motion are reduced to a homogeneous quasilinear system of partial differential equations of mixed type. The flow parameters in the intermediate mixed layer are determined from the equilibrium conditions in a more general model of three-layer flow of a miscible fluid. In particular, the equilibrium conditions imply the constancy of the interlayer Richardson number in velocity-shift flows. A self-similar solution to the problem of breakdown of an arbitrary discontinuity (the lock-exchange problem) in the domain of hyperbolicity of the system under consideration is constructed. The transcritical flow regimes over a local obstacle are studied and the conditions under which the obstacle determines the upstream flow are determined. A comparison of steady-state and time-dependent solutions with the solutions obtained for the original three-layer models of miscible fluid flow is carried out.

Exact self-similar solutions for radiative heat transfer equations in a kinetic model with regular mode

Exact self-similar solutions for radiative heat transfer equations in a kinetic model with regular mode

On the solvability of a space-time fractional nonlinear Schrödinger system

This paper is devoted to the theoretical analysis of a coupled nonlinear system of fractional Schrödinger equations in Rn×R+,n≥1, considering time fractional derivative in the Caputo sense, and a fractional spatial dispersion defined in terms of the Fourier transform. We prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, considering power-type nonlinearities and initial data in the framework of weak-Lp spaces, which contain singular functions with infinite energy. As consequence of the embedding of weak-Lp spaces in Lloc2, for p>2, the obtained solutions have finite local L2-mass. In addition, we discuss the scenario in which it is possible to obtain the existence of self-similar solutions, which is a symmetric property that reproduces the structure of physical phenomena in different spatio-temporal scales. Our results are applicable, in the fractional setting, to the nonlinear Schrödinger and Biharmonic equations, as well as in a large class of dispersive systems appearing in nonlinear optics.

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

Super-Eddington Magnetized Neutron Star Accretion Flows: A Self-similar Analysis

The properties of super-Eddington accretion disks exhibit substantial distinctions from the sub-Eddington ones. In this paper, we investigate the accretion process of a magnetized neutron star (NS) surrounded by a super-Eddington disk. By constructing self-similar solutions for the disk structure, we study in detail an interaction between the NS magnetosphere and the inner region of the disk, revealing that this interaction takes place within a thin boundary layer. The magnetosphere truncation radius is found to be approximately proportional to the Alfvén radius, with a coefficient ranging between 0.34–0.71, influenced by the advection and twisting of a magnetic field, NS rotation, and radiation emitted from an NS accretion column. Under super-Eddington accretion, the NS can readily spin up to become a rapid rotator. The proposed model can be employed to explore the accretion and evolution of NSs in diverse astrophysical contexts, such as ultraluminous X-ray binaries or active galactic nucleus disks.

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

Jetsimpy: A Highly Efficient Hydrodynamic Code for Gamma-Ray Burst Afterglow

Gamma-ray burst (GRB) afterglows are emissions from ultrarelativistic blast waves produced by a narrow jet interacting with surrounding matter. Since the first multimessenger observation of a neutron star merger, hydrodynamic modeling of GRB afterglows for structured jets with smoothly varying angular energy distributions has gained increased interest. While the evolution of a jet is well described by self-similar solutions in both ultrarelativistic and Newtonian limits, modeling the transitional phase remains challenging. This is due to the nonlinear spreading of a narrow jet to a spherical configuration and the breakdown of self-similar solutions. Analytical models are limited in capturing these nonlinear effects, while relativistic hydrodynamic simulations are computationally expensive, which restricts the exploration of various initial conditions. In this work, we introduce a reduced hydrodynamic model that approximates the blast wave as an infinitely thin two-dimensional surface. Further assuming axial symmetry, this model simplifies the simulation to one dimension and drastically reduces the computational costs. We have compared our modeling to relativistic hydrodynamic simulations and semianalytic methods, and applied it to fit the light curve and flux centroid motion of GRB 170817A. These comparisons demonstrate good agreement and validate our approach. We have developed this method into a numerical tool, jetsimpy, which models the synchrotron GRB afterglow emission from a blast wave with arbitrary angular energy and Lorentz factor distribution. Although the code is built with GRB afterglow in mind, it applies to any relativistic jet. This tool is particularly useful in Markov Chain Monte Carlo studies and is provided to the community.

Self-Similar Solutions for the Membrane Transverse Vibration Equation

Self-Similar Solutions for the Membrane Transverse Vibration Equation

Impact of disc magnetisation on MHD disc wind signature

Context. Observations of blue-shifted X-ray absorption lines indicate the presence of wind from the accretion disc in X-ray binaries. Magnetohydrodynamic (MHD) driving is one possible wind-launching mechanism. Recent theoretical developments have made self-similar magnetic accretion-ejection solutions much more generalised, showing that wind can be launched at a much lower magnetisation than the equipartition value, which had previously been the only possibility. Aims. In this work, we model the transmitted spectra through MHD-driven photoionised wind models with different levels of magnetisation. We investigate the possibility of detecting absorption lines by upcoming instruments, such as XRISM and Athena. We investigate the robustness of the method of fitting asymmetric line profiles by multiple Gaussians. Methods. We used the photoionisation code XSTAR to simulate the transmitted model spectra. To cover the extensive range of velocity and density of the wind spanned over a large distance (∼105 gravitational radii), we divided the wind into slabs following a logarithmic radial grid. Fake observed spectra are finally produced by convolving model spectra with instrument responses. Since the line asymmetries are apparent in the convolved spectra as well, this can be used in future XRISM and Athena spectra as an observable diagnostic to fit for. We applied some amount of rigor in assessing the equivalent widths of the major absorption lines, including the Fe XXVI Lyα doublets, which will be clearly distinguishable thanks to the superior quality of future high-resolution spectra. Results. Disc magnetisation stands as another crucial MHD variable that can significantly alter the absorption line profiles. Pure MHD outflow models at low magnetisation are dense enough to be observed by the existing or upcoming instruments. Therefore, these models can serve as simpler alternatives to MHD-thermal models. Fitting with multiple Gaussians is a promising method for handling asymmetric line profiles, as well as the Fe XXVI Lyα doublets.

Nonlinear tunneling of self-similar periodic waves in inhomogeneous two mode optical fibers

Distinct types of self-similar periodic waveforms of the generalized coupled nonlinear Schrödinger equations with varying nonlinearity, gain or loss and group velocity dispersion are obtained. The coupled system applies to the description of light pulse propagation in an inhomogeneous two mode optical fiber. The self-similar solutions are expressed in terms of Jacobian elliptic functions, thus enabling us to identify various kinds of propagating self-similar soliton pulses in their long-wave limit. It is found that these periodic structures exhibit a linear chirp property, which can be utilized to achieve efficient pulse compression and amplification. As a physically relevant application, we discuss the nonlinear tunneling process of these linearly chirped self-similar periodic waves through both dispersion and nonlinear barriers.

Self-similar collapse in Painlevé–Gullstrand coordinates

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

The forward self-similar solution of fractional incompressible Navier-Stokes system: The critical case

In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli,Kohn and Nirenberg-type iteration, L∞ estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [28] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.

Correction to: Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

Correction to: Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

Jet formation model from accretion disks of electron-ion-photon gas

The problem of Astrophysical Jet formation from relativistic accretion disks through the establishment of relativistic disk-powerful jet equilibrium structure is studied applying the Beltrami-Bernoulli equilibrium approach of Shatashvili and Yoshida 2011; Arshilava et al. 2019. Accretion disk is weakly magnetized consisting of fully ionized relativistic electron-ion plasma and photon gas strongly coupled to electrons due to Thompson Scattering. Analysis is based on the generalized Shakura-Sunyaev α-turbulent dissipation model for local viscosity (being the main source of accretion), in which the contributions from both the photon and ion gases are taken into account. Ignoring the self-gravitation in the disk we constructed the analytical self-similar solutions for the equilibrium relativistic disk-jet structure characteristic parameters in the field of gravitating central compact object for the force-free condition. It is shown, that the magnetic field energy in the Jet is several orders greater compared to that of accretion disk, while jet-outflow is locally Super-Alfvénic with local Plasma-beta <1 near the jet-axis. The derived solutions can be used to analyze the astrophysical jets observed in binary systems during the star formation process linking the jet properties with the parameters of relativistic accretion disks of electron-ion-photon gas.

Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$ , the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$ . Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$ . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.