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An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coefficient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a finite number of variables. A similar result is true for Stefan's problem. In the limit, when the number of phases tends to infinity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function. [ABSTRACT FROM AUTHOR]

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer's function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay. [ABSTRACT FROM AUTHOR]

When studying the boundary value problems' solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the three-dimensional parabolic-elliptic Patlak-Keller-Segel chemotactic model. The initial data is almost a Dirac measure supported on a straight line with mass less than 8 π. We prove that if the data is sufficiently close to the straight line, then global well-posedness holds. The method in this paper is a refinement of the work on vortex filament solutions of the Navier-Stokes equations by Bedrossian, Germain and Harrop-Griffiths [1]. [ABSTRACT FROM AUTHOR]

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

The author studies the 2D isentropic Euler equations with the ideal gas law. He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time 1 3 -Hölder bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. The proof of the solutions, following the 3D construction of Buckmaster, Shkoller and Vicol (in 2023), is based on the stable 2D self-similar Burgers profile and the modulation method. [ABSTRACT FROM AUTHOR]

We develop a weak noise approximation for the Kolmogorov forward equation governing the dynamics of a leaky integrate-and-fire neuron subject to white noise. Although being very simple, our approximation provides accurate results as far the magnitude of noise-induced fluctuations Δ remains much smaller than the distance A between the mean potential (center of mass) and the excitation threshold. The error for the firing rate is 3 for the stationary stimuli and if A/Δ>5 for time-varying stimuli.

In this paper, we study the one-phase Stefan problem on a surface of revolution, characterizing those surfaces admitting a self-similar solution. [ABSTRACT FROM AUTHOR]

Particle dissolution is an important process that occurs during heat treatment in the transportation process. We present a one-dimensional model of spherical wax particle dissolution in a fluid. The model includes a nonlinear equation at the free boundary, which was established to describe the dissolution rate of a wax particle assuming a diffusion process. At the moving boundary, we compute the concentration distributions of partial solutions. To estimate wax dissolution in solution and related expressions for the dissolved radius, an analytical solution of a mathematical model for spherical particle dissolution in flow is used. [ABSTRACT FROM AUTHOR]

The problem of determining the characteristics of a laminar swirling wall jet of an incompressible fluid is considered. Numerical solutions of the Navier–Stokes equations are obtained in the stationary and non-statitonary formulations. It is shown that the jet characteristics obey a self-similar law at large distances from the jet source, as in the case of a three-dimensional laminar non-swirling jet, but in our case the jet propagates at a certain angle to the initial direction of jet blowing. With a large swirling of flow in the jet, regions of recirculation flow appear and the jet flow becomes unsteady. [ABSTRACT FROM AUTHOR]

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature, and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs. [ABSTRACT FROM AUTHOR]

In this article, we consider the Riemann problem with arbitrary initial data for one-dimensional blood flow equations in the arterial circulation. Here, we establish the existence of the self-similar solution to the Riemann problem by limiting viscosity approach. We convert the Riemann problem to a boundary value problem by adding a suitable viscosity term and establish the existence of the solution. Finally, we construct the existence of the solution to this Riemann problem in the presence of vacuum state. [ABSTRACT FROM AUTHOR]

We study a one-dimensional system of equations for a discrete gas model (the McKean system). The McKean system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two groups of particles. Under certain conditions on a singularity variety, the system passes the Painlevé test. In addition, the kinetic system admits a reduction to a system of ordinary differential equations for which the Painlevé test is performed and it becomes possible to find a solution. [ABSTRACT FROM AUTHOR]

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Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type , Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions. [ABSTRACT FROM AUTHOR]

Assessment of pipeline safety is of paramount significance for safety of petrochemical plants. Any leakage of flammable gas from a pipeline poses a potential risk of an explosion followed by fire. Following a leak, a non-burning inert gaseous jet is first formed and, thereafter, ignition happens and an exothermal torch or flame occur. Safety assessments of such a scenario can be made by considering free-boundary-layer turbulent flows of fuel propagating in the surrounding atmosphere based on Prandtl's mixing length theory. That implies modeling the fields of fuel/oxidizer concentrations, velocity, and temperature, and the overall shape of the axisymmetric inert jets and exothermal diffusion torches. Axisymmetric turbulent boundary layer equations have been solved numerically, and these solutions have been compared with self-similar solutions for inert jets. The comparison confirms that the difference between the numerical and self-similar solutions in the longitudinal velocity profile is within 1%. Furthermore, the numerical model predicts the fuel concentration within an uncertainty level of 15% when compared with the experimental data. In an exothermal torch, the combustion temperature keeps increasing until the fuel is depleted. Outside the fuel zone, the temperature of the torch cools abruptly in the radial direction. The longitudinal velocity of an exothermal torch is moderately higher than that of an inert jet because of the thermal expansion. In this work, results of parametric studies are presented for various values of the Reynolds number, which can be useful for petrochemical engineers who assess the risk level associated with potential fire hazards caused by pipeline leakages. [ABSTRACT FROM AUTHOR]

We study the dynamics of unsaturated flows within a fluid-infused wet patch in a thin porous layer, coupled with the transport of passive particles, partly inspired by some art practice and environmental and industrial activities. The role of wetting and capillary forces and the influence of evaporation are both emphasized. Two coupled partial differential equations (PDEs) are first derived to describe the time evolution of the fluid saturation and concentration of passive particles. Three dimensionless parameters are identified after appropriate nondimensionalization of the coupled PDEs, the influence of which is demonstrated through asymptotic and numerical solutions at both the early and late times. With evaporation, in particular, the fluid front first arrives at a maximum location after an initial period of spreading. Thereafter, the front starts to recede until the fluid drains out at a finite time. Correspondingly, the concentration of particles blows up at a finite time, when the size of the wet region shrinks to zero. Meanwhile, the spatial distribution of particles evolves towards a universal curve near the front at late times, as the particles gradually accumulate at the front of the wet patch. It is recognized that, while the spreading of fluid within a porous layer behaves similar to that of a viscous gravity current subject to volume loss, the frontal structure for the concentration of particles, to some extent, is similar to that of an intruding inviscid gravity current. [ABSTRACT FROM AUTHOR]

Based on the mechanics of multiphase media methods and equations, a mathematical model of superheated water vapor injection into a reservoir initially saturated with methane is constructed. It is assumed that water vapor condensation occurs at frontal boundary. Considered problem is reduced to solving the problem of finding two moving boundaries: the boundary of vapor condensation and the boundary of methane displacement by water. Self-similar solutions of the problem are obtained that describe the pressure and temperature distributions in the reservoir. [ABSTRACT FROM AUTHOR]

This paper is concerned with two aspects of the fractional Navier-Stokes equation. First, we establish the local L2 theory of the hypo-dissipative Navier-Stokes system. More precisely, the existence of local in time as well as global in time local energy weak solutions to the hypo-dissipative Navier-Stokes system is proved. In particular, in order to construct a pressure with an explicit representation, some technical innovations are required due to the lack of known results on the local regularity of the non-local Stokes operator. Secondly, as an important application to the local L2 theory, we give a second construction of large self-similar solutions of the hypo-dissipative Navier-Stokes system along with the Leray-Schauder degree theory. [ABSTRACT FROM AUTHOR]

The article deals with a boundary value problem of mathematical physics describing the unsteady process of diffusion of a reacting gas in the direction of a spherical coal particle located in the atmosphere of a mine. The solution of the boundary value problem is based on self-similar transformations which are a special case of group analysis. Formulas for determining the concentration of the reacting gas in the vicinity of the coal particle and on its surface are obtained. Graphs of the dependencies of the burn-out time of the coal particle on a number of its parameters are constructed, and the fields of the reacting gas concentration at various stages of burning of the coal particle are revealed. [ABSTRACT FROM AUTHOR]

There are the exact solutions to nonautonomous evolution equation with two space variables the right side of which contains the Monge–Ampere operator. The solutions with additive and multiplicative separation of variables are found. The reductions of the given equation to ordinary differential equations (ODEs) are considered. The classic and generalized self-similar solutions and the solutions with functional separation of variables are obtained. In particular, it is shown that the given equation can be reduced to ODE if the coefficient at the time derivative can be represented in the form of product of the functions depending on time, space variables, and sought function. The reductions of the given equation to two-dimensional partial differential equations are also determined. [ABSTRACT FROM AUTHOR]

In this paper, we obtained sufficient and necessary conditions for self-similar motions of three point vortices in generalized two-dimensional fluid systems by reducing of Hamiltonian and developing the theory of self-similar functions. The results are concise and clear, and the occurrence of both self-similar collapse and self-similar expansion only depend on the strengths of point vortices. We have provided an explicit and exact expression for each nontrivial self-similar solution. The application of theoretical results in Surface-Quasi-Geostrophic system is consistent with previous known numerical results in the symmetric case. We also have found that, in some asymmetric cases, both self-similar collapse and self-similar expansion occur if and only if the strength of the third point vortex is within an open interval. • Established sufficient and necessary conditions for self-similar motions. • Provided explicit expressions for nontrivial self-similar solutions. • Defined criteria for self-similar collapse and expansion in vortex systems. • The theory of self-similar functions is developed. [ABSTRACT FROM AUTHOR]

In this article, we study the long-time asymptotic properties of a non-linear and non-local equation of diffusive type which describes the rock–paper–scissors game in an interconnected population. We fully characterize the self-similar solution and then prove that the solution of the initial–boundary value problem converges to the self-similar profile with an algebraic rate. [ABSTRACT FROM AUTHOR]

We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, such as can model micro-electro-mechanical systems. For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal aymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin quenches at the origin like an explicit spatially independent solution. The latter analysis and numerical experiments suggest a detailed conjecture for the singular behaviour. [ABSTRACT FROM AUTHOR]

This paper examines nonlinear partial differential equation (PDE) solutions. Scientists and engineers have struggled to solve nonlinear differential equations. Nonlinear equations arrive in nearly all problems in nature. There are no well-established techniques for solving all nonlinear equations, and efforts have been made to enhance approaches for a specific class of problems. Keeping this in mind, we shall investigate the perturbation method's efficiency in solving nonlinear PDEs. Several techniques work well for diverse issues. We recognize that there may be several solutions to a given nonlinear issue. Methods include homotropy analysis, tangent hyperbolic function, factorization and trial function. However, some of these strategies do not cover all nonlinear issue solutions. In this paper, we use the perturbation technique to solve the zeroth-order Airy equation and also find the Bessel function in the first-order nonhomogeneous differential equation by using self-similar solutions that appears in modified Korteweg–de Vries (KdV) equation. This approach will be used for nonlinear equations in physics and applied mathematics. [ABSTRACT FROM AUTHOR]

In this paper, a fluid model is presented which contains the general linear equation of state including the gravitation term. The obtained spherical symmetric Euler equation and the continuity equations were investigated with the Sedov-type time-dependent self-similar ansatz which is capable of describing physically relevant diffusive and disperse solutions. The result of the space and time-dependent fluid density and radial velocity fields are presented and analyzed. Additionally, the role of the initial velocity on the kinetic and total energy densities of the fluid is discussed. This leads to a model, which can be considered as a simple model for a dark-fluid. [ABSTRACT FROM AUTHOR]

A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of di®erential equations. Further, the solution is constructed using the Painlevé series. Here the system already passes the Painlevé test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level. [ABSTRACT FROM AUTHOR]

We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in the Cartesian coordinate system. We couple the two fluids through the continuity equation where different void fractions can be considered. The well-known self-similar Ansatz was applied and analytic solutions were derived for both velocity and pressure field as well.

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of self-similar analytical solutions to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary for polytropic gases. It is shown that the free boundary will grow linearly in time (see (1.11)), moreover, both angular velocity u ϕ and its derivatives, and the derivative of radial velocity u r will tend to zero as t → + ∞ , while radial velocity itself is bounded. In particular, it is interesting to see that when the adiabatic exponent γ > 2 and for a suitably large time, the rotation plays a dominant role in accelerating the growth of the free boundary compared with the effect of pressure. [ABSTRACT FROM AUTHOR]

The review is devoted to exact analytical solutions for quasi-2D gravity segregated flows or gravity currents in subterranean porous formations. The problems under consideration are quasi-linear. The driving forces are two components of the buoyancy—one exerting the bulk of the light fluid and one due to the curvilinearity of the interface between the fluids. In the case of homogeneous formation or where the seal slope is negligible, the transport equation is parabolic and allows for a wide set of self-similar solutions. In a large-scale approximation of the buoyancy domination, the governing equation is hyperbolic; the method of characteristics allows for a detailed analytical description of gravity current propagation with final accumulation in the geological trap. Analytical models for leakage via the caprock seal are also discussed. The work was completed by formulating some unsolved problems in segregated flows in porous media. [ABSTRACT FROM AUTHOR]

The results of an analytical study of unsteady heat transfer in the vicinity of a stagnation point of an axisymmetric impinging liquid jet are presented. A self-similar solution is obtained, which allows the analysis of heat transfer behavior as a function of the Reynolds number, distance from the nozzle to the surface, and characteristic time. The behavior of the Nusselt number for the solution in asymptotically limiting cases is shown: the Froude number Fr ≫ 1 (small distances from the nozzle to the barrier) and Fr ≪ 1 (large distances). It is noted that the effect of the Froude number can be quite significant and should be taken into account when predicting heat transfer. Significant intensification of heat transfer is shown for a pulsed impinging jet at short pulse durations. [ABSTRACT FROM AUTHOR]

A partially invariant solution of a three-dimensional problem with a free boundary for the Navier–Stokes equations is studied. The flow domain under consideration is a horizontal layer bounded by a solid plane from below and by a flat free surface from above. The vertical velocity and pressure are independent of the and coordinates. Three flow modes can be formed for different initial velocities of the flow: stabilization to the quiescent state with time, solution blow up within a finite time, and self-similar regime in which the layer thickness unboundedly increases with time. [ABSTRACT FROM AUTHOR]

Studying self-similar solutions of geometric flows on manifolds plays an important role in understanding geometrical and topological properties of underlying manifolds. In this paper, we prove that a complete shrinking Ricci–Bourguignon harmonic flow soliton ((M , g) , (N , h) , ϕ , X , ρ , λ) is compact if and only if | | X | | is bounded on M. Also, we show that a complete shrinking Ricci–Bourguignon harmonic flow soliton has finite fundamental group. [ABSTRACT FROM AUTHOR]

Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained. [ABSTRACT FROM AUTHOR]

In the article on the basis of the equation of motion of the compressible lubricant for the «thin layer», continuity, state, and the equation describing the profile of the molten contour taking into account the formula of mechanical energy dissipation the asymptotic and self-similar solution is found for the extreme (when the speed tends to infinity) and non-extreme case. As a result of solving the problem, a refined mathematical calculation model of a wedgeshaped sliding support with a low-melting metal coating on a movable contact surface is obtained, which compensates for an emergency lack of lubricant and provides a stable hydrodynamic lubrication mode.

In this study we examine the classical problem of fluid flow in an aquifer, obeying the transient Darcy–Forchheimer law. This problem is solved by using the symmetry properties of the governing equations (e.g. the mass balance equation and the transient Darcy–Forchheimer momentum equation) which enable us to transform the time and the space coordinates into one independent coordinate. According to our study the flow in the aquifer may be divided into two main components. One component is the steady part of the flow discharge and the other one is the transient part of the flow discharge. The obtained solution shows that the reduction in the above-mentioned transient part leads to the creation of three zones: (1) the “near zone” located near the inlet face to the aquifer and is characterized by a positive flow; (2) the “far zone” in the aquifer lying at an infinite distance from the inlet face where the flow is also positive and (3) the “intermediate backflow zone”, which is lying among the above-mentioned zones and is characterized by reverse flow. The results obtained in this study may be useful for understanding the transient flow process in the aquifer, stemming from the Darcy–Forchheimer flow, and for the prediction of the discharge and head distribution in the aquifer. [ABSTRACT FROM AUTHOR]

We suggest new methods of solving boundary value problems for some classical equations of the laminar boundary layer theory, namely, an heuristic method of reducing a nonlinear boundary value problem for the Falkner–Skan equation to a linear Cauchy problem and a group-theoretic approach to integrating the Blasius equation describing the motion of an incompressible flow near a semi-infinite plate. [ABSTRACT FROM AUTHOR]

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdélyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problems for ordinary differential equation. An essential role is played by certain properties of Erdélyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data. [ABSTRACT FROM AUTHOR]

The expansion problem of ideal gas and incompressible spherical monodisperse particles mixture in a closed shock tube has been posed and solved. An asymptotically accurate solution of the initial (self-similar) stage of expansion of the gas suspension with a fine powder fraction was obtained. For the nonequilibrium case of larger particles, the problem was solved numerically by the hybrid large-particle method of the second-order approximation in space and time. The double-speed splitting effects of the flow stratification and the interface both into the gas-phase contact and the porosity jump as well as the interaction of the reflected shock wave with them were revealed. For the late stage of expansion, the dynamics of the gas suspension layer was established to be similar to a nonlinear oscillatory dissipative system with attenuation and a drift of the split contact interfaces depending on the size of dispersed inclusions.

In this paper, we consider a family of closed hypersurfaces which shrink self‐similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres. [ABSTRACT FROM AUTHOR]