Time Asymptotics Of Solutions Research Articles

Stability of asymptotic waves in the Fisher–Stefan equation

We establish spectral, linear, and nonlinear stability of the vanishing and slow-moving travelling waves that arise as time asymptotic solutions to the Fisher–Stefan equation. All stability analysis is in terms of the limiting equations that the asymptotic waves satisfy.

The large time asymptotic solutions of nonlinear Schrödinger type equations

We give a short description of the proof of asymptotic-completeness for NLS-type equations with radial data in three dimensions. We also show some aspects of the method by giving a new proof of Asymptotic Completeness for the two-body Quantum Scattering case.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

On Solutions to a Class of Functional Differential Equations with Time-Dependent Coefficients

In this paper, we study initial boundary value problems that involve functional (nonlocal) partial differential equations with variable coefficients. These problems arise in cell growth models with symmetric and asymmetric modes of division. We determine the general solution to the symmetric cell division problem for a certain class of coefficients and establish the convergence of solutions to a large time asymptotic solution. The existence of a steady size distribution (SSD) solution for an asymmetric cell division problem is established and is shown to be the large time-attracting solution for a certain class of coefficients. The rate of convergence of solutions towards the SSD solution is affected by the choice of coefficients and remains unaffected by the asymmetry in cell division. The uniqueness of solutions to the initial boundary value problem is also established.

A Near-surface Temperature Model of Arrokoth

A near-surface thermal model for Arrokoth is developed based on the recently released 105 facet model of the body. This thermal solution takes into account Arrokoth’s surface reradiation back onto itself. The solution method exploits Arrokoth’s periodic orbital character to develop a thermal response using a time-asymptotic solution method, which involves a Fourier transform solution of the heat equation, an approach recently used by others. We display detailed thermal solutions assuming that Arrokoth’s near-surface material’s thermal inertia 2.5 W/m−2 K−1 s1/2. We predict that at New Horizons’ encounter with Arrokoth, its encounter hemisphere surface temperatures were ∼57–59 K in its polar regions, 30–40 K in its equatorial zones, and 11–13 K for its winter hemisphere. Arrokoth’s orbitally averaged temperatures are around 30–35 K in its polar regions and closer to 40 K near its equatorial zones. Thermal reradiation from the surrounding surface amounts to less than 5% of the total energy budget, while the total energy ensconced into and exhumed out of Arrokoth’s interior via thermal conduction over one orbit is about 0.5% of the total energy budget. As a generalized application of this thermal modeling together with other Kuiper Belt object origins considerations, we favor the interpretation that New Horizons’ REX instrument’s 29 ± 5 K brightness temperature measurement is consistent with Arrokoth’s near-surface material being made of sub-to-few-millimeter-size tholin-coated amorphous H2O ice grains with 1 W/m−2 K−1 s1/2 10–20 W/m−2 K−1 s1/2 and which are characterized by an X-band emissivity in the range 0.9 and 1.

Analytical solutions for one-dimensional diabatic flows with wall friction

Abstract

Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles

Abstract

A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: Numerical analysis and exploration

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the equations exhibit diversified solutions from various numerical observations, the properties of solutions are not yet completely understood, and by far there has been no rigorous numerical analysis work concerning such models. We propose a conservative and conditionally positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models.

Asymptotics for the fourth-order nonlinear Schrödinger equation in 2D

Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrödinger equation in two space dimensions [Formula: see text] where [Formula: see text] We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition [Formula: see text] We continue to develop the factorization techniques. The crucial points of our approach presented here are the [Formula: see text] — estimates of the pseudodifferential operators and the application of the Kato–Ponce commutator estimates.

Existence and Uniqueness of a Non-Negative Monotonic Solution of a Nonlinear Ordinary Differential Equation

In this paper, we prove the existence and uniqueness of a non-negative monotonically increasing solution of the boundary value problem (BVP) $$\begin{aligned}&H''(\eta )-H^p(\eta )H'(\eta )+\frac{\eta }{2}H'(\eta )+\frac{1}{2p}H(\eta )=0,\quad \eta >0,\\&H(0)=0,\quad \lim _{\eta \rightarrow +\infty }\eta ^{-\frac{1}{p}}H(\eta )=1, \end{aligned}$$ using a shooting argument for $$p>1$$ . This BVP arises when we construct the large time asymptotic solution of the modified Burgers equation, using the method of matched asymptotic expansions, on the quarter plane.

Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.

Foam-liquid front motion in Eulerian coordinates.

A mathematical model formulated as a system of Hamilton-Jacobi equations describes implicitly the propagation of a foam-liquid front in an oil reservoir, as the zero-level set of the solution variable. The conceptual model is based on the 'pressure-driven growth' model in Lagrangian coordinates. The Eulerian mathematical model is solved numerically, where the marching is done via a finite volume scheme with an upwind flux. Periodic reinitialization ensures a more accurate implicit representation of the front. The numerical level set contour values are initially formed to coincide with an early time asymptotic analytical solution of the pressure-driven growth model. Via the simulation of the Eulerian numerical model, numerical data are obtained from which graphical representations are generated for the location of the propagating front, the angle that the front normal makes with respect to the horizontal and the front curvature, all of which are compared with the Lagrangian model predictions. By making this comparison, it is possible to confirm the existence of a concavity in the front shape at small times, which physically corresponds to an abrupt reorientation of the front over a limited length scale.

Hydraulic fracturing with leakoff in a pressure-sensitive dual porosity medium

Hydraulic fracturing is a key method in the stimulation of shale gas reservoirs. Shale gas formations often contain natural fractures which are fluid-pressure sensitive and dilate in response to the inflation of the fracture, increasing fluid loss and slowing down and potentially prematurely arresting fracture propagation. Models typically assume 1-D single-porosity/permeability (Carter) leakoff perpendicular to the hydraulic fracture. However, the leakoff process in naturally fractured formations is considerably more complex. In this study, we present an hydraulic fracturing model based on the PKN-formalism which accommodates leakoff into a pressure-sensitive dual porosity medium. Proppant transport is accommodated by introducing empirical constitutive equations to determine the proppant distribution during the hydraulic fracturing treatment. The model is solved numerically and is validated against known small and large time asymptotic solutions. The model is capable of providing a rapid estimation of the morphology of hydraulic fractures in naturally fractured formations and the corresponding proppant distribution. The simulation results illustrate that the leakoff into a dual porosity medium, where fracture permeability is a strong function of applied fluid pressure, results in a reduced length of the propagating fracture due to the fugitive fluid leakoff from the fracture into the surrounding formation and that this in turn results in a reduced maximum width during the treatment. The ability to infuse proppants in fluid-driven fractures penetrating large distances from the injection wellbore is further limited by premature screen-out. This may compromise the ultimate efficiency of the final hydraulic fracture regarding gas recovery. Reduced propagation and premature screen-out are limited by low permeability and large spacing of the natural fractures. The presence of an existing network of natural fractures, including those adjacent to the hydraulic fracture that may become propped, aids in the recovery of the resource by reducing diffusion lengths of the hydrocarbon to the main fracture.

Wave breaking for the Stochastic Camassa–Holm equation

We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.

Factorization technique for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the cubic fourth-order nonlinear Schrodinger equation $$\left\{\begin{array}{ll}i\partial_{t}u+\frac{1}{4}\partial_{x}^{4}u = i \lambda \partial _{x}(\left| u \right| ^{2}u),&\quad t > 0,\, x \in \mathbf{R},\\ u \left( 0,x\right) = u_{0}\left( x\right) ,&\quad x \in \mathbf{R},\end{array}\right.$$ where \({\lambda \in \mathbf{R}.}\) We introduce the factorization formula for the free evolution group to prove the global existence of solutions. Also we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.

Slope modulation of ring waves governed by two-dimensional sine–Gordon equation

A theory of slope modulation of waves governed by the two-dimensional sine–Gordon equation is proposed. A large time asymptotic solution describing the slope modulation of trains of ring kinks is obtained. The comparison with the numerical solution of two-dimensional sine–Gordon equation shows excellent agreement.

Large Time Asymptotics for the Reduced Ostrovsky Equation

We consider the Cauchy problem for the reduced Ostrovsky equation $$u_{tx} = u + \left(u^{3}\right)_{xx}$$ with real valued initial data \({u \left(0\right) = u_{0}}\). We introduce the factorization for the free evolution group to prove the global existence of solutions. Also, we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.

Three-dimensional finite element method for rotating disk flows

This paper deals with the numerical simulation of rotating disk flow, coupling the hydrodynamic field to the transport of a chemical species generated in electrochemical cells by the dissolution of an iron electrode in a 1M H2SO4 electrolyte. The time asymptotic steady-state solution of rotating disk flow obtained in this work by numerical integration of the incompressible Navier–Stokes equations through the finite element method (FEM) with the appropriate boundary conditions is consistent with the generalized von Karman’s similarity solution. This paper reviews the main features of generalized von Karman’s flow, the adopted FEM scheme, a discussion of implementation of the boundary conditions, validation of the code and the results. In particular, the results confirm the assumption that the non-dimensional velocity and concentration profiles do not depend on the coordinate along the radial direction.

Self-similar asymptotics for linear and nonlinear mathematical models of tumor angiogenesis: a review

We show that the long time asymptotic solutions of initial value problems for linear and nonlinear mathematical models of tumor angiogenesisare self-similar spreading solutions. The symmetries of the governing equationsyield three-parameter families of these solutions given in terms of their mass,center of mass, and variance. Unlike the mass and center of mass, the variance,or ”time-shift,” of a solution is not a conserved quantity for the non linear problem

Extended Stokes' flows in cylindrical geometries

Stokes' flows of an isochoric, Newtonian fluid in cylindrical geometries are analytically investigated. Transient and time asymptotic solutions are deduced and their main features as well as applications to engineering problems are discussed. In the classical problems a circular cylinder translates along its symmetry axis or rotates around it, the axial or azimuthal wall speed behaving in time as a finite step or periodically. The resulting velocities in the fluid filling the outside or the inside of the cylinder and the wall stresses involve Macdonald's functions (external flows) or modified Bessel functions of the first kind (internal) of order 0 or 1. Extended azimuthal and axial Stokes' problems are also introduced and solved. In the azimuthal problems, the cylindrical wall is cut in two parts by a plane normal to the axis: one part rotates, while the other one is kept at rest. The behavior of the azimuthal velocities and of the stresses in a neighborhood of the above plane is discussed. In the axial problems a strip (or also a finite number of strips) of the cylindrical wall translates, while its remaining part is kept at rest. Velocities and wall stresses are obtained by means of azimuthal Fourier series involving Macdonald's or modified Bessel functions of any integral order.