Asymptotic Solutions of a First-Order Differential Equation with Deviating Argument and Slowly Varying Coefficients

The object of this paper is to study the problem of constructing an approximate solution of a first-order weakly nonlinear ordinary differential equation with deviating argument and slowly varying coefficients. On the basis of asymptotic techniques in nonlinear mechanics, we construct an algorithm for the asymptotic integration of the differential equation under consideration.__________Published in Neliniini Kolyvannya, Vol. 7, No. 4, pp. 475–486, October–December, 2004.     

Global Continua of Positive Solutions for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition

This paper is concerned with a quasilinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.     

Strong Traces for Solutions to Scalar Conservation Laws with General Flux

In this paper we consider bounded weak solutions u of scalar conservation laws, not necessarily of class BV, defined in a subset . We define a strong notion of trace at the boundary of reached by L ¹ convergence for a large class of functionals of u, G(u). The functionals G depend on the flux function of the conservation law and on the boundary of . The result holds for a general flux function and a general subset.     

Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.     

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behavior of non-Newtonian fluids in engineering sciences is often modeled by a power law ansatz with p ∈ (1, 2]. In the present paper the local in time existence of strong solutions is studied. The main result includes also the degenerate case (δ = 0) of the extra stress tensor and thus improves previous results of [L. Diening and M. Růžička, J. Math. Fluid Mech., 7 (2005), pp. 413–450].     

Existence of Global Entropy Solutions of a Nonstrictly Hyperbolic System

In this paper we use the theory of compensated compactness coupled with some basic ideas of the kinetic formulation by Lions, Perthame, Souganidis & Tadmor [LPS, LPT] to establish an existence theorem for global entropy solutions of the nonstrictly hyperbolic system (1).     

Structure of Solutions of Multidimensional Conservation Laws with Discontinuous Flux and Applications to Uniqueness

We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.     

Two-phase Entropy Solutions of a Forward–Backward Parabolic Equation

This article deals with the Cauchy problem for a forward–backward parabolic equation, which is of interest in physical and biological models. Considering such an equation as the singular limit of an appropriate pseudoparabolic third-order regularization, we consider the framework of entropy solutions, namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation. Moreover, we restrict the attention to two-phase solutions, that is solutions taking values in the intervals where the parabolic equation iswell-posed, proving existence and uniqueness of such solutions.     

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

The generalized form of the well-known Mathieu differential equation, which consists of two driving force terms, including the quadratic and cubic nonlinearities, has been analyzed in this paper. The two-dimensional Lindstedt–Poincarés perturbation technique has been considered in order to obtain the analytical solutions. The transition curves in some special cases have been presented. It is shown that the periodic solution does indeed exist and in general they are dependent on the initial conditions. Results of this analytical approach were compared with those obtained from the numerical methods and it is found that they are in a good agreement.     

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.     

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.     

Large Time Behavior for a Quasilinear Diffusion Equation with Critical Gradient Absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption

$$\begin{aligned} \partial _t u-\Delta _{p}u+|\nabla u|^{q_*}=0 \quad \hbox {in}\, (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

for \(p\in (2,\infty )\) and \(q_*:=p-N/(N+1)\). We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the p-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

     

Existence of 3D Strong Solutions for a System Modeling a Deformable Solid Inside a Viscous Incompressible Fluid

We study a coupled system modeling the movement of a deformable solid inside a viscous incompressible fluid. For the solid we consider a given deformation which has to obey several physical constraints. The motion of the fluid is modeled by the incompressible Navier–Stokes equations in a time-dependent bounded domain of \(\mathbb {R}^3\), and the solid satisfies the Newton’s laws. Our contribution consists in adapting and completing in dimension 3, some existing results, in a framework where the regularity of the deformation of the solid is limited. We rewrite the main system in domains which do not depend on time, by using a new means of defining a change of variables, and a suitable change of unknowns. We study the corresponding linearized system before setting a local-in-time existence result. Global existence is obtained for small data, and in particular for deformations of the solid which are close to the identity.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

Periodic Step-Like Contrast Structures for a Quasilinear Parabolic Equation with Small Diffusion Coefficient

Using the method of boundary functions, for a quasilinear parabolic equation with small diffusion coefficient we construct an asymptotic expansion of a periodic solution with internal transition layer. Sufficient conditions for the existence of this solution are obtained. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 329–350, July–September, 2005.     

Exponential Dichotomy and Time-Bounded Solutions for First-Order Hyperbolic Systems

The paper is devoted to studying a class of strongly hyperbolic systems of the first order. We show that if the characteristic roots of the full symbol are outside an open strip containing the real axis, then the homogeneous system possesses an exponential dichotomy and the inhomogeneous system is solvable in the space of time-bounded and almost periodic functions. We also discuss some results on the behavior of solutions for nonlinear equations in the neighborhood of a stationary point.     

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.     

On Asymptotic Properties of Continuously Differentiable Solutions of Quasilinear Functional Differential Equations

We establish new properties of C ¹ [τ(1), + ∞)-solutions of the quasilinear functional differential equation