Global smooth solutions for a quasilinear fractional evolution equation

The global existence of smooth solutions to a class of quasilinear fractional evolution equations is proved. The proofs are based on L^p(L^q) L^p(L^q) maximal regularity results for the corresponding linear equations.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

Measures on phase space as solutions of the one-dimensional neutron transport equation

Summary The linear integral transport operator for slab geometry is formulated and studied as a mapping on the set of measures on the phase space of the underlying system, with the expected number of neutrons emergent from a collision represented by a measure on the space of outgoing velocities. Under appropriate assumptions it is shown that, if c represents the maximum number of secondary particles per collision, then there exists c ₁ ≥1 such that the system is subcritical for c≤c ₁ . An example shows that c ₁ ≥1 is sharp in general, but further assumptions are given under which one can deduce c ₁ >1. The idealized laws of elastic and inelastic scattering are shown to satisfy our assumptions. Entrata in Redazione il 27 ottobre 1975.     

Global smooth solutions of IBVP to nonlinear equation of suspended string

We shall consider IBVP to a nonlinear equation of suspended string with uniform density to which a nonlinear time-independent outer force works. The nonlinear term is smooth but not monotone. We shall show that IBVP has a unique time-global smooth solution. The regularity of the solutions shall be also studied.     

New solutions for the one-dimensional nonconservative inviscid Burgers equation

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac-δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp localised oscillations. In the present paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This will be afforded by our theory of distributional products, and is based on a rigorous and consistent concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641-656]. Our approach exhibit Dirac-δ travelling solitons (they are just the “infinitesimal narrow solitons” of Maslov, Omel'yanov and Tsupin [V.P. Maslov, O.A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981) 73-149; V.P. Maslov, V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dokl. 24 (1979) 354-356]) and also solutions which are not measures such as for instance u(x,t)=b+δ^′(x−bt), a wave of constant speed b. Moreover, for signals with two jump discontinuities we have, in our setting, the propagation of more solitons and more values for the signal speed are allowed than those afforded within classical framework.     

Global smooth solution for the compressible Landau-Lifshitz-Bloch equation

The Landau-Lifshitz-Bloch equation is often used to describe micromagnetic phenomenon under high temperature. In this paper, we establish the existence and uniqueness of global smooth solution for the initial problem of the compressible Landau-Lifshitz-Bloch equation in dimension one.     

Global regular solutions for Landau-Lifshitz equation

In this note, we prove that there exists a unique global regular solution for multidimensional Landau-Lifshitz equation if the gradient of solutions can be bounded in space L ²(0, T; L ^∞). Moreover, for the two-dimensional radial symmetric Landau-Lifshitz equation with Neumann boundary condition in the exterior domain, this hypothesis in space L ²(0, T; L ^∞) can be cancelled.     

Global one-dimensional optimization using smooth auxiliary functions

In this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has Lipschitzean first derivatives. To solve this problem, smooth auxiliary functions, which are adaptively improved during the course of the search, are constructed. Three new algorithms are introduced: the first used the exact a priori known Lipschitz constant for derivatives; the second, when this constant is unknown, estimates it during the course of the search and finally, the last method uses neither the exact global Lipschitz constant nor its estimate but instead adaptively estimates the local Lipschitz constants in different sectors of the search region during the course of optimization. Convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Global solutions to the eikonal equation

We study structural stability of smoothness of the maximal solution to the geometric eikonal equation on (R^d,G)$({\mathbb{R}}^d,G)$, d?2$d?2$. This is within the framework of order zero metrics G. For a subclass of these metrics we show existence, stability as well as precise asymptotics for derivatives of the solution. These results are applicable to examples arising in Schrödinger operator theory.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

Global solutions for a one-dimensional problem in conducting fluids

A mathematical model of the motion of conducting fluids is studied in this paper. The dynamics of such fluids is described by the equations of compressible fluids coupled to the Maxwell’s equations. We prove global existence of strong solution for a one-dimensional initial-boundary value problem of this model (plane conducting flows) with general large data.     

Global dynamics above the ground state energy for the one-dimensional NLKG equation

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein–Gordon (NLKG) equation on the line with focusing nonlinearity ${|u|^{p-1}u, p >5 }$ , provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case (Nakanishi and Schlag in Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, preprint, 2010), the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein–Gordon equation, similar to those of Mizumachi (J Math Kyoto Univ 48(3):471–497, 2008). The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.     

Global smooth solution of hydrodynamical equation for the Heisenberg paramagnet

In this note, we obtain the existence and uniqueness of global smooth solution for the Cauchy problem of multidimensional hydrodynamical equation for the Heisenberg paramagnet. Copyright © 2004 John Wiley & Sons, Ltd.     

Conservative similarity solutions of the one-dimensional autonomous parabolic equation

Summary conditions are established on the coefficient functions of the one-dimensional autonomous parabolic equation for the existence of conservative solutions in the first quadrant of the plane of the independent variables length and time. Such a solution is characterized by the condition that its integral over the length variable from zero to infinity exist and be independent of time. Those conservative solutions are considered which can be expressed by means of a dimensionless variable and a function of the time variable (similarity). For the case that the similarity relation is linear in the length variable, coefficient functions are specified that lead to conservative solutions.

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Zusammenfassung Es werden Bedingungen aufgestellt über die Koeffizientenfunktionen der eindimensionalen, autonomen, parabolischen Gleichung für die Existenz konservativer Lösungen imersten Quadranten der Ebene der unabhängigen Variablen Länge und Zeit. Eine derartige Lösung ist dadurch charakterisiert, dass ihr Integral über die Längenvariable von Null bis Unendlich existiert und unabhängig von der Zeit ist. Es werden konservative Lösungen betrachtet, die mit Hilfe einer dimensionslosen Variablen ausgedrückt werden könen, die als Quotient einer Funktion der Längenvariablen und einer Funktion der Zeitvariablen definiert ist (Ähnlichkeit). Koeffizientenfunktionen, die zu konservativen Lösungen führen, werden für den Fall angegeben, dass diese Ähnlichkeitsbeziehung linear in der Längenvariablen ist.