Quasilinear Initial Boundary Value Problems Research Articles

Weakly Nonlinear Multiphase Geometric Optics for Hyperbolic Quasilinear Boundary Value Problems: Construction of a Leading Profile

We investigate in this paper the existence of the leading profile of a WKB expansion for quasilinear initial boundary value problems with a highly oscillating forcing boundary term. The framework is weakly nonlinear, as the boundary term is of order O(e) where the frequencies are of order O(1/e). We consider here multiple phases on the boundary, generating a countable infinite number of phases inside the domain, and we therefore use an almost periodic functional framework. The major difficulties of this work are the lack of symmetry in the leading profile equation and the occurrence of infinitely many resonances (opposite to the simple phase case studied earlier) The leading profile is constructed as the solution of a quasilinear problem, which is solved using a priori estimates without loss of derivatives. The assumptions of this work are illustrated with the example of isentropic Euler equations in space dimension two.

Strong solution of a hydrodinamics problem with memory

In the last few years, the concepts of fractional calculus were frequently applied to other disciplines. Recently, this subject has been extended in various directions such as signal processing, applied mathematics, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics. In fluid dynamics, the fractional derivative models were used widely in the past for the study of viscoelastic materials such as polymers in the glass transition and in the glassy state. Recently, it has increasingly been seen as an efficient tool through which a useful generalization of physical concepts can be obtained. The fractional derivatives used most are the Riemann--Liouville fractional derivative and the Caputo fractional derivative. It is well known that these operators exhibit difficulties in applications. For example, the Riemann--Liouville derivative of a constant is not zero. We deal with so called temporal fractional derivative as a prototype of general fractional derivative. We prove the global strong solvability of a linear and quasilinear initial-boundary value problems with a singular complete monotone kernels. Our main tool is a theory of evolutionary integral equations. An abstract fractional order differential equation is studied, which contains as particular case the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. This paper concerns with an initial-boundary value problem for the Navier--Stokes--Voigt equations describing unsteady flows of an incompressible viscoelastic fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution in two-dimensional domain. We also establish an $L_2$ decay estimate for the velocity field under the assumption that the external forces field is conservative.

The Lifespan for Nonlinear Wave Equation Outside of Star-Shaped Obstacle in Three Space Dimensions

In this paper, we consider the initial-boundary value problem of quasi-linear wave equation outside of star-shaped obstacle in three space dimensions. We study the general case that the nonlinear term may explicitly depend on the unknown function itself. By some new weighted space-time L 2 estimates of the linear wave equation for the unknown function itself, we obtained sharp lower bounds of the lifespan for the quasi-linear initial boundary value problem.

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

AbstractThis is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and different type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

AbstractIn this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and different type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

AbstractThis is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and different type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.

Convergence of finite element approximation to quasilinear initial boundary value problems

Convergence of finite element approximation to quasilinear initial boundary value problems