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We consider for a small parameter > 0 a parabolic convection–diffusion problem with Péclet number of order ( − 1) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order () connected through a domain (node) of diameter (). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as → 0 , i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems (= 0) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed. [ABSTRACT FROM AUTHOR]

This paper deals with the initial value problem for a semilinear wave equation on a bounded domain and solutions are required to vanish on the boundary of this domain. The essential feature of the situation considered here is that the ellipticity of the spatial part of the differential operator degenerates like the square of the distance from given point in the domain and so hyperbolicity is lost at this point. The assumptions ensure that u ≡ 0 is a stationary solution of the problem and the object is to study the stability of this solution with respect to perturbations of the initial data. Stability, instability and asymptotic stability are all considered. The assumptions about the nonlinear terms ensure that the problem has a well-defined linearization at u ≡ 0. There are simple cases where this linearization is asymptotically stable but u ≡ 0 is an unstable solution of the nonlinear problem. We also establish conditions under which the stability of a stationary solution u ≢ 0 can be determined using our results. The quadratic degeneracy at a point treated here is typical of what is required in models for acoustic (or sonic) black holes. It also occurs in a simplified Wheeler–DeWitt model which we discuss in some detail. [ABSTRACT FROM AUTHOR]

We study the well-posedness and the long-time behavior of almost periodic solutions to stochastic degenerate parabolic–hyperbolic equations in any space dimension, under the assumption of Lipschitz continuity of the flux and viscosity functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problems of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation. [ABSTRACT FROM AUTHOR]

The document is an author index for Volume 20 of the Journal of Hyperbolic Differential Equations. It lists the authors and titles of articles published in the volume, along with the page numbers and publication dates. The articles cover a range of topics in hyperbolic differential equations, including wave equations, conservation laws, and fluid dynamics. The index provides a useful resource for library patrons conducting research in these areas. [Extracted from the article]

In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears. [ABSTRACT FROM AUTHOR]

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line [ 0 , + ∞) for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line [ 0 , + ∞) by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval [ 0 , 1 ] by introducing some new average quantities. [ABSTRACT FROM AUTHOR]

Practically speaking, fluid flow in round and noncircular nozzle is a very regular occurrence. The cold and hot water used in our homes is delivered to us via pipes. Water is delivered throughout the city via large pipe networks. Large pipelines carry natural gas and oil hundreds of kilometres from one place to another. During the operation of an engine, cooling water is carried by hoses to the radiator's pipes, in which it is cooled as it travels. Experimentally, we detected the results of the model because there is no restriction on the application of experimental research to a certain sector or kind of concept. It may be used for a broad range of events and circumstances. Under parabolic velocity conditions, fluid (Water Pr 6.9) flows from the inlet position. The top and bottom walls of the rectangular nozzle are also moving at the same velocity as they are at the inlet position. Due to the movement of the walls, fluid is compressed in the particular region and also exhibits the same parabolic behavior. The solution of the coupled equations is determined by using the Finite Volume Method (FVM). When partial differential equations are expressed as algebraic equations, the FVM may be used to evaluate them. It can be used to evaluate elliptic, parabolic, and hyperbolic partial differential equations. Using FVM, it is necessary to know the values (and derivatives) of multiple variables at the cell faces, when the values (and derivatives) of these variables are only known at the cell centres. When determining these variables for convective terms, it is common to take the direction of the flow into consideration. The numerical results of the velocity and the pressure could be seen in the rectangular nozzle. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots τ j verify an inequality like τ j 2 (x) + τ k 2 (x) ≤ M (τ j (x) − τ k (x)) 2. We prove that the Cauchy problem is well-posed in ∞ if the operators with frozen coefficients are uniformly hyperbolic in the sense of Gårding. [ABSTRACT FROM AUTHOR]

In this work, we describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call "pressure") which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem. [ABSTRACT FROM AUTHOR]

We derive and analyze the linearized hyperbolic equations describing a relativistic heat-conducting elastic rod. We construct a decreasing energy integral for these equations, compute the associated characteristic propagation speeds and prove that the solutions decay in time by using a Fourier decomposition. For comparison purposes, we obtain analogous results for the classical system with heat waves, in which the finite propagation speed of heat is kept but the other relativistic terms are neglected and also for the usual classical system. [ABSTRACT FROM AUTHOR]

We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability. [ABSTRACT FROM AUTHOR]

We extend the Erdős-Rényi law of large numbers to the averaging setup both in discrete and continuous time cases. We consider both stochastic processes and dynamical systems as fast motions whenever they are fast mixing and satisfy large deviations estimates. In the continuous time case we consider flows with large deviations estimates which allow a suspension representation and it turns out that fast mixing of corresponding base transformations suffices for our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435-457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249-275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235-292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266] to cover those complicated models. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed. [ABSTRACT FROM AUTHOR]

We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity. [ABSTRACT FROM AUTHOR]

We consider a nearly hyperbolic Sasakian manifold equipped with -structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with -structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with -structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with -structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with -structure under the condition when f is parallel. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions or, more generally, nonconstant monotonic bounded functions as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multi-type version of the sticky particle dynamics and we obtain the existence of global weak solutions via a compactness argument. We then derive a stability estimate on the particle system which is uniform in the number of particles. This allows us to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161(1) (2005) 223-342]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all order, which extends the classical estimate and generalizes to diagonal systems a result by Bolley, Brenier and Loeper [Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ. 2(1) (2005) 91-107] in the scalar case. Our results are established without any smallness assumption on the variation of the data, and we only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic. [ABSTRACT FROM AUTHOR]

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map of the polydisc which does not admit fixed points in . We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic and the Abel equation for a parabolic nonzero-step . This is done by studying the canonical Kobayashi hyperbolic semi-model of and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation, we also describe the space of all solutions. [ABSTRACT FROM AUTHOR]

A third-order entropy stable scheme for nonlinear hyperbolic conservation laws is proposed here. This scheme contains two main ingredients: a fourth-order entropy conservative flux and a third-order numerical diffusion operator. A piecewise-quadratic reconstruction from pointwise values is developed in order to approximate the third-order dissipative term. To guarantee a non-oscillating property, a nonlinear limiter is employed and, furthermore, the scheme is proven to be entropy stable. Finally, numerical experiments are presented and demonstrate the accuracy, high-resolution, and robustness of our method. [ABSTRACT FROM AUTHOR]

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl. 1 (2010) 341-376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations 248 (2010) 1972-2002]. [ABSTRACT FROM AUTHOR]

We study properties of evolution equations which are first order in time and arbitrary order in space (FTNS). Following Gundlach and Martín-García (2006) we define strong and symmetric hyperbolicity for FTNS systems and examine the relationship between these definitions, and the analogous concepts for first-order systems. We demonstrate equivalence of the FTNS definition of strong hyperbolicity with the existence of a strongly hyperbolic first-order reduction. We also demonstrate equivalence of the FTNS definition, up to N = 4, of symmetric hyperbolicity with the existence of a symmetric-hyperbolic first-order reduction. [ABSTRACT FROM AUTHOR]

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505-527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357-1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235-277] for systems of hyperbolic conservation laws without source terms. [ABSTRACT FROM AUTHOR]

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints. [ABSTRACT FROM AUTHOR]

We are interested in nonlinear partial differential equations of hyperbolic type, written as first-order balance laws posed on a curved spacetime, whose unknown is typically a section of the bundle of future-oriented, timelike vectors. We are especially interested in dealing with systems and solutions with low regularity and, to this end, we introduce here the notion of "strictly convex entropy field". We then prove that a system of balance laws endowed with such an entropy admits a symmetrization which, however, may be "degenerate" if the entropy is not uniformly convex or if the constitutive laws defining the balance laws have limited regularity. Next, we establish a weak-strong uniqueness and stability theorem for the initial value problem associated with balance laws endowed with a strictly convex entropy field. We compare two solutions with limited regularity: on the one hand, a continuous solution that need not be Lipschitz continuous and, on the other hand, a weak solution that satisfies an "entropy inequality". Finally, we apply our theory to the Euler system for compressible fluid flows on a curved spacetime, and we exhibit an entropy field, which is strictly convex but fails to be uniformly convex as vacuum is approached. This leads us to a uniqueness theorem for (both the relativistic and non-relativistic versions of) the Euler system, which applies to a continuous solution with vacuum and an entropy weak solution with vacuum. [ABSTRACT FROM AUTHOR]

A family of convex solutions of Φxx - f(x)Φyy = 0, for x > 0 and y ∈ ℝ, where f is positive and continuously differentiable in (0, ∞), is discussed. It consists of all convex solutions of that equation which are of the form Φ(x, y) = p(x)q(y). The separation of variables is an easy task to perform. In particular, it results in an explicit form of q(y). Imposing convexity conditions requires however more insight. It is observed that a nonlinear part of those conditions, in case of f′ ≤ 0, is related to an asymptotic behavior of p(x) and p′(x) as x → ∞. Then, under an additional assumption that x→∞ f(x) > 0, a satisfactory description of the set of all the functions p(x), which determines convex Φ(x, y) via the formula Φ(x, y) = p(x)q(y), is obtained. So constructed functions Φ(x, y) are convex entropies for the corresponding p-system. Finally two nontrivial examples, involving a modified Bessel and hypergeometric equation are provided. [ABSTRACT FROM AUTHOR]

The Muthuswamy-Chua system describes the simplest electronic circuit which can have chaotic phenomena. In this paper, we first prove the existence of three families of consecutive periodic orbits of the system when α = 0, two of which are located on consecutive invariant surfaces and form two invariant topological cylinders. Then we prove that for α > 0 if the system has a periodic orbit or a chaotic attractor, it must intersect both of the planes z = 0 and z = -1 infinitely many times as t tends to infinity. As a byproduct, we get an example of unstable invariant topological cylinders which are not normally hyperbolic and which are also destroyed under small perturbations. [ABSTRACT FROM AUTHOR]

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in ℙ³ of degree d ≥ 5. We also show that there exist Kobayashi hyperbolic surfaces in ℙ³ of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Our method yields also some lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d ≥ 15 in ℙ4. [ABSTRACT FROM AUTHOR]

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear stochastic partial differential equations (spde's) of parabolic and hyperbolic type. These equations rely on a spatial operator given by the L2-generator of a d-dimensional Lévy process X = (Xt)t≥0, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index H > 1/2. As an application, we consider the case when X is a β-stable process, with β ∈ (0, 2]. In the parabolic case, we develop a connection with the potential theory of the Markov process (defined as the symmetrization of X), and we show that the existence of the solution is related to the existence of a "weighted" intersection local time of two independent copies of . [ABSTRACT FROM AUTHOR]

We study a mixed type equation, which is analogous, in parts, to Tricomi equation and Keldysh equation. For this equation, the characteristics in the hyperbolic region are transversal to the transition locus in a subset of the locus, but is tangential to it in another subset. We propose a formulation of a closed boundary value problem for such a mixed type equation, and prove that it admits a unique H1 solution with some higher regularity inside the domain. Our result shows the variety and the complexity of the boundary value problems for mixed-type equations. [ABSTRACT FROM AUTHOR]

We are concerned with the stability of steady solutions to the initial and boundary value problem associated with reaction-hyperbolic systems for axonal transport. Under proper structural assumptions, we clarify the relaxation structure of the reaction-hyperbolic systems and establish the large-time asymptotic stability of steady solutions and relaxation boundary-layers. [ABSTRACT FROM AUTHOR]

This paper studies a hyperbolic modification of Maxwell's equations to ensure Gauss' law. This correction was obtained by adding a parameter-dependent new unknown and is of great interest for the numerical simulation in plasma physics since the discretization of the Maxwell-Vlasov system does not grant straightforwardly the physical conservation of the charge. Such problems are encountered while using Particle-In-Cell schemes. In this paper the new proposed system has the interest of still being a Friedrichs' one. Its asymptotic behavior with respect to the parameter and the link between modified and original Maxwell's systems are thus investigated. At last, we look for some boundary conditions, granting the well-posedness of the system. Generalizations of the Silver-Müller condition, perfect electric and magnetic conductors, as well as impedance and admittance representation of materials are detailed. [ABSTRACT FROM AUTHOR]

We consider a class of second-order hyperbolic systems which describe viscoelastic materials, and we extend the recent results by Dharmawardane and Conti et al. More precisely, for all initial data (u0, u1)∈(Hs+1(ℝN) ∩ L1, γ(ℝN))×(Hs(ℝN) ∩ L1, γ(ℝN)) with γ∈[0, 1], we derive faster decay estimates for both dissipative structure or regularity-loss type models. To this end, we first transform our problem into Fourier space and then, by using the pointwise estimate derived by Dharmawardane et al., combined with a device to treat the Fourier transform in the low frequency region, we derive optimal decay results for the solutions to our problem. Finally, we use these decay estimates for the linear problem, combined with the weighted energy method introduced by Todorova and Yordanov, and tackle a semilinear problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the reflection of nonlinear discontinuous waves, for weakly well-posed hyperbolic boundary value problems, satisfying the (WR) condition, that is in a case where the IBVP is neither strongly stable, nor strongly unstable. We study how the singularities of a striated solution are reflected when the solution hits the boundary. We prove striated estimates and L∞ estimates and observe the loss of one derivative: we show that a discontinuity of the gradient of the solution across a hyperplane can be reflected in a discontinuity across a hyperplane of the solution itself. [ABSTRACT FROM AUTHOR]

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation $\partial_{t}^2u = u\partial_x(u \partial_x u)$, which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude. [ABSTRACT FROM AUTHOR]

For hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa. [ABSTRACT FROM AUTHOR]

We derive a quadratically nonlinear equation that describes the motion of a tangential discontinuity in incompressible magnetohydrodynamics near the onset of a Kelvin-Helmholtz instability. We show that nonlinear effects enhance the instability and give a criterion for it to occur in terms of a longitudinal strain of the fluid motion along the discontinuity. [ABSTRACT FROM AUTHOR]

We consider Maxwell equations for an electromagnetic field propagating in carbon nanotubes (CNTs) placed on a dispersive nonmagnetic dielectric medium. We obtain the effective equation analogous to the classical sine-Gordon equation. Then it has been analyzed numerically. We have revealed the dependence of the pulse on the type of CNT and on the initial pulse amplitude, as well as on the medium dispersion constants. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-Riemannian quantity H in Finsler geometry. We obtain some rigidity theorems of a compact Finsler manifold under some conditions related to H. We also prove that the S-curvature for a Randers metric is almost isotropic if and only if H almost vanishes. In particular, S-curvature is isotropic if and only if H = 0. [ABSTRACT FROM AUTHOR]

For the p-system of fluid dynamics and under the assumption that the initial data have small oscillation but not necessarily small total variation, we reproduce the so-called Glimm-Lax estimates with greater accuracy. Our approach recovers the results known for smooth flux functions, and is based on the theory of generalized characteristics and Dafermos' technique of polygonal approximations. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-linear hyperbolic system, occuring in the study of electromagnetic fields in the presence of superconductors. The constitutive relation between current density and the electric field is then highly non-linear. Based on a stability estimate in the dual space, we are able to prove the convergence of backward Euler's method toward the unique solution of the problem. This requires a compensated compactness argument (div-curl lemma) and the Minty-Browder procedure to pull weak convergence through a monotone non-linearity. Finally, we present the corresponding error estimates. [ABSTRACT FROM AUTHOR]

We consider a discrete kinetic approximation of the isentropic Euler equations, and establish the local convergence of the solutions of these relaxation systems to those of the hydrodynamic equations in the hyperbolic limit. We rely on modulated entropy methods and cover the time interval in which the latter admits smooth solutions. [ABSTRACT FROM AUTHOR]

We study the generalized Tricomi problem for the Lavrentiev-Bitsadze equation in a sector, when the boundary condition prescribed in the hyperbolic region is far away from the characteristic. The existence and uniqueness of a solution to this problem is proven and further estimates of interest are established. [ABSTRACT FROM AUTHOR]

We propose a reformulation of ideal magnetohydrodynamics written in the Lagrangian variable as an enlarged system of hyperelastic type, with a specific potential. We study the hyperbolicity of the model and prove that the acoustic tensor is positive for all directions which are non orthogonal to the magnetic field. The consequences for Eulerian ideal magnetohydrodynamics and for numerical discretization are briefly discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solution of quasilinear hyperbolic equations with linear damping. The main novelty lies in the following observation: If we suitably choose the initial data of the corresponding parabolic equation, then the solution Ψ = Ψ(x, t) of the parabolic equation served as the new asymptotic profile satisfies |(V-Ψ, (V-Ψ)x, (V-Ψ)t)(t)|L∞ = O(1)(t-2, t-5/2, t-3). The convergence rates of the new profile Ψ are better than that obtained by H.-J. Zhao (2000, J. Differential Equations167, 467-494), and we need none of the additional technical assumptions (H1) and (H2) therein. Therefore, we answer an open problem posed by Nishihara (1997, J. Differential Equations133, 384-395). [ABSTRACT FROM AUTHOR]

We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be "weakly" well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth. [ABSTRACT FROM AUTHOR]