Applications of Schochet''s methods to parabolic equations

Methods used by S. Schochet in [32] enable one to find a lower bound for the life span of solutions of hyperbolic PDEs with a small parameter. We prove a similar theorem for such equations where a diffusion term has been added, with the minimal assumption on the Sobolev regularity of the initial data ( in the d-dimensional torus). When the data is smooth and under a “small divisor” assumption on the perturbation, the first term of an asymptotic expansion of the solution is computed. Those results are then applied to prove global existence theorems, for arbitrary initial data, in the case of the primitive system of the quasigeostrophic equations, followed by the rotating fluid equations. We finally prove a more precise existence theorem for the latter, using anisotropic Sobolev and Besov spaces.     

Two-Step Fifth-Order Methods for Evolutionary Problems with Positive Operators

The family of three-level fifth-order time integrators are considered for hyperbolic, parabolic PDEs and stiff ODEs. They are classified into two parts depending on positivity or negativity of the operators corresponding to the second time derivatives. Two options are presented for hyperbolic equations. Stability analysis is performed for linear cases. It is shown that the scheme for ODEs can be nearly A-stable and nearlyL -stable for particular values of its free parameter. Numerical illustration is presented for hyperbolic case.     

Construction and analytic properties of integral manifolds of solutions of nonlinear systems of difference equations

We study properties of integral manifolds of a system of difference equations in the hyperbolic case. We prove the existence of analytic integral manifolds for a system of difference equations with analytic right sides. We examine an analytic dependence on the parameter.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 630–636, May, 1992.     

On construction of weak solutions with higher integrable gradients to linear hyperbolic partial differential systems

Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic differential equations. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 30–52.     

Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω ? limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

Lp – Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper L_p – L_q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.     

Parametric optimization for a hyperbolic equation in divergence form with a pointwise state constraint: I

We consider an optimal control problem with a pointwise state constraint of inequality type and with dynamics described by a linear hyperbolic equation in divergence form with the homogeneous Dirichlet boundary condition. The state constraint contains a functional parameter that belongs to the class of continuous functions and occurs as an additive term. We study the properties of solutions of linear hyperbolic equations in divergence form with measures in the input data and compute the first variations of functionals on the basis of a so-called two-parameter needle variation of controls.     

Parabolic approximations of diffusive–dispersive equations

We consider a lower-order approximation for a third-order diffusive–dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive–dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive–dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive–dispersive law. The proofs rely on new a priori energy estimates for higher-order derivatives and the technique of compensated compactness.     

On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C ^k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.     

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schläfli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schläfli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schläfli's differential formula, as consequences of scissors congruence relations.     

Modulation equations and parabolic limits of reaction random-walk systems

Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

This paper presents an MLP‐type neural network with some fixed connections and a backpropagation‐type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with n equations and n unknowns. The proposed structure is based on a backpropagation‐type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Maxwell's equations when the charge conservation is not satisfied

Maxwell's equations are overdetermined when the charge conservation equation is not verified. In order to overcome this problem, different methods have been introduced. We notice that they fit into a framework in which a new formulation which we introduce also fits. These formulations can be classified according to the type of the resulting PDE-system as hyperbolic-elliptic, hyperbolic-parabolic and purely hyperbolic. We show that the resolution of Maxwell's equation through the potentials is always equivalent to the purely hyperbolic formulation and that the hyperbolic-parabolic and hyperbolic-elliptic formulations converge to the purely hyperbolic formulation when introducing a parameter which goes to 0.     

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

On the Riemann matrices of the hyperbolic system dual to the system of equations of one-dimensional gas dynamics

We consider the Cauchy problem for the quasilinear hyperbolic system describing a one-dimensional flow of a gas with the equation of state p = p(ϱ), p′(ϱ) > 0, and with initial data satisfying a monotonicity condition. We suggest an approach to solving it by reduction to the Cauchy problem for the linear hyperbolic system obtained from the original system by the hodograph transformation. These constructions are extended to a system of elasticity equations describing nonlinear vibrations of a one-dimensional medium. The main result is illustrated by two examples.     

A new closure mitigating the sub-shock phenomenon in the continuous shock-structure problem

Moment equations offer an attractive framework for modeling non-equilibrium processes in moderately rarefied gases. We present a new hyperbolic closure for the 5-moment system in the one-dimensional setting that couples the conservation laws of mass, momentum and energy to two evolution equations of higher, non-equilibrium moments. The new closure, given in closed-form, depends on a regularization parameter β that is related to the non-linearity around the equilibrium state, so as to mitigate the appearing sub-shock in the continuous shock-structure problem. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the coastal ocean over a time period of several weeks

From a scale analysis of hydrodynamic phenomena having a significant action on the drift of an object in coastal ocean waters, we deduce equations modeling the associated hydrodynamic fields over a time period of several weeks. These models are essentially non linear hyperbolic systems of PDE involving a small parameter. Then from the models we extract a simplified and nevertheless typical one for which we prove that its classical solution exists on a time interval which is independent of the small parameter. We then show that the solution weak-∗ converges as the small parameter goes to zero and we characterize the equation satisfied by the weak-∗ limit.     

On the existence and nonexistence of solutions of some quasilinear hyperbolic equations

We consider a special class of quasilinear hyperbolic equations of arbitrary order suggested by V.A. Galaktionov. For these equations, we prove the existence of solutions periodic in t > 0 and consider an initial-boundary value problem for which we derive sufficient conditions for the nonexistence of a global solution in the natural energy space of solutions.