Representations for the solutions and coefficients of evolution equations

We give new representations for the solutions and coefficients of evolution equations in the linear case. The obtained formulas contain some functional arbitrariness that can be used in identification problems. We also give classes of hyperbolic equations that admit the generalized functionally invariant solutions.     

Variational methods for fluid–structure interaction and porous media

In this work we consider a poroelastic, flexible material that may deform largely, which is situated in an incompressible fluid driven by the Navier–Stokes equations in two or three space dimensions. By a variational approach we show existence of weak solutions for a class of such coupled systems. We consider the unsteady case, this means that the PDE for the poroelastic solid involves the Fréchet-derivative of a non-convex functional as well as (second order in time) inertia terms.     

On a class of nonconvex and nonlinear optimal control problems

We propose two relaxation approaches for the existence of solutions of a nonconvex optimal control problem with a nonlinear dynamics and two fixed endpoints. The first approach adds to the functional a term depending on the final state; the second one introduces a new scalar control into both the functional and the dynamics. Our results generalize those obtained in the linear case. We assume that the integrand be concave w.r.t. the state variable, and provide an example showing that a strict concavity condition, in the nonlinear case, is essential. Finally, a result without relaxation is presented.     

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

In this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in 3 .     

On the asymptotics of the motion of a nonlinear viscous fluid

Under study is the nonstationary problem of the motion of a nonlinear-viscous fluid in the case of low or high viscosity. We establish that the convergence of solutions to the corresponding limit solutions as the viscosity converges to zero or infinity.     

New algebraic approaches to classical boundary layer problems

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.     

Blow-up and global solutions to a new integrable model with two components

We will discuss a new integrable model which describes the motion of fluid. The present work is mainly concerned with global existence and blow-up phenomena which are largely due to the application of conservation laws for this integrable equations. Moreover, a new blow-up criterion for nonperiodic case is also established via the associated potential. Some interesting examples are also given to illustrate the application of our results. The precise blow-up rate is also investigated. Finally, we will emphasize the relations of classical Camassa-Holm equation and our model by analyzing the existence of global solutions.     

Discontinuous solutions to the Euler equations for a van der Waals fluid

In this work we study the discontinuous solutions to the Euler equations for a van der Waals fluid, which contain one shock and one phase transition. We consider the general case when there is a characteristic between the shock front and the phase boundary. We establish the local existence of such solutions.     

Third-order partial differential equations arising in the impulsive motion of a flat plate

We obtain numerical solutions to a class of third-order partial differential equations arising in the impulsive motion of a flat plate for various boundary data. In particular, we study the case of constant acceleration of the plate, the case of oscillation of the plate, and a case in which velocity is increasing yet acceleration is decreasing. We compare the numerical solutions with the known exact solutions in order to establish the validity of the method. Several figures illustrating both solution forms and the relative strength of the second and third-order terms are presented. The results obtained in this study reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena.     

Numerical solution of the free‐surface viscous flow on a horizontal rotating elliptical cylinder

The numerical solution of the free‐surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time‐dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free‐surface is determined as part of the solution. For the time‐dependent case a simple time‐marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break‐down occurs. Time‐dependent solutions do not produce the periodic solution after a long time‐scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub‐maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Classification of the types of instability of vertical flows in geothermal systems

Stability of vertical flows in geothermal systems is investigated in the case when the domain occupied by water (heavy fluid) is located over the domain occupied by vapor. It is found that under the transition to an unstable regime in a neighborhood of the existing solution, a pair of new solutions appears as a result of the turning point bifurcation. We consider the dynamics of a narrow band of weakly unstable and weakly nonlinear perturbations of the plane surface of the water-to-vapor phase transition. It is shown that such perturbations obey the generalized Ginzburg-Landau-Kolmogorov-Petrovsky-Piscounov equation.     

Global attractor for a low order ODE model problem for transition to turbulence

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy‐conserving, non‐quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Applications of <Emphasis Type="Italic">p</Emphasis>-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions

In a very general setting, we discuss possibilities of applying p-adics to geophysics using a p-adic diffusion representation of the master equations for the dynamics of a fluid in capillaries in porous media and formulate several mathematical problems motivated by such applications. We stress that p-adic wavelets are a powerful tool for obtaining analytic solutions of diffusion equations. Because p-adic diffusion is a special case of fractional diffusion, which is closely related to the fractal structure of the configuration space, p-adic geophysics can be regarded as a new approach to fractal modeling of geophysical processes.     

Conversion of second-class constraints and resolving the zero-curvature conditions in the geometric quantization theory

In the approach to geometric quantization based on the conversion of second-class constraints, we resolve the corresponding nonlinear zero-curvature conditions for the extended symplectic potential. From the zero-curvature conditions, we deduce new linear equations for the extended symplectic potential. We show that solutions of the new linear equations also satisfy the zero-curvature condition. We present a functional solution of these new linear equations and obtain the corresponding path integral representation. We investigate the general case of a phase superspace where boson and fermion coordinates are present on an equal basis.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.     

Similarity Solutions to the Stefan Problem and the Binary Alloy Problem

We present a catalogue of explicit similarity solutions to theStefan problem and the binary alloy problem; the most generalcase includes convective heat transfer due to a fluid motiondriven by a density change at the solid-liquid interface. Thefree boundary is either an ellipsoid or a hyperboloid; in thelatter case we note that for certain parameter regimes theremay be two solutions (with different initial data), a situationwhich does not occur in the one-dimensional case.     

On the three-dimensional finite Larmor radius approximation: The case of electrons in a fixed background of ions

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows, Han-Kwan (2010) [18], where the three-dimensional analysis of a Vlasov–Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell–Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in Han-Kwan (2010) [18]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive from the Vlasov–Poisson equation a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the inverse of the intensity of the magnetic field and the Debye length vanish) to a new anisotropic fluid system. This is achieved thanks to Cauchy–Kovalevskaya type techniques, as introduced by Caflisch (1990) [7] and Grenier (1996) [14]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.     

A multidimensional nonlinear sixth-order quantum diffusion equation

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.     

A new entropy functional for a scalar conservation law

In this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics. The generalization has two aspects: The new entropy functional is defined not for one but for two solutions. It is defined in terms of the L₁ distance between the two solutions as well as the variations of each separate solution. In addition, it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero. © 1999 John Wiley & Sons, Inc.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.