Compensated Compactness Method Research Articles

Global solutions for the modified Camassa–Holm equation

AbstractThis paper is concerned with the global solutions for the modified Camassa–Holm equation. We first obtain the global existence and uniqueness of strong solutions for the initial datum with , and then give some norm estimates of strong solutions and its momentum density. Finally, by using the compensated compactness method and regularization technique, we establish the global existence and uniqueness of weak solutions for the initial datum .

Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas–Fermi equations. We derive approximate solutions by the fractional step Lax–Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas–Fermi equations for weak entropy–entropy flux pairs, generated by compactly supported C0∞ test functions, are confined in a compact set in Hloc−1. We prove that the Young measure must be a Dirac measure by the Tartar–Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.

Investigation of Aerodynamic Characteristics of Aircrafts in a Rarefied Gas Flow Using the Moment Method

A one-dimensional nonstationary nonlinear moment system of equations and an approximation of Maxwell’s microscopic boundary condition will be introduced. The flight speed and surface temperature of the aircraft are included in the moment system of equations as coefficients. Macroscopic boundary conditions also depend on the surface temperature of the aircraft. The quantity of macroscopic boundary conditions for the moment system of equations depends on the parity of the approximation number of the moment system of equations. We state the initial and boundary value problem for the moment system of equations in the third approximation under macroscopic boundary conditions. This paper proves the existence and uniqueness of the solution of the abovementioned problem in the space of functions continuous in time and summable in the square by spatial variable. The theorem on the existence and uniqueness of a solution of the initial and boundary value problem for the moment system of equations in the third approximation is proved by the method of a priori estimation and using the Galerkin method and Tartar’s compensated compactness method.

Singular limits of the quasi-linear Kolmogorov-type equation with a source term

Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic–parabolic (ultra-parabolic) equation with a smooth source term is established. In addition, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is investigated and the formal limit is rigorously justified. Our proofs rely on the use of kinetic equations and the compensated compactness method for genuinely nonlinear balance laws.

Weak and strong semigroups in structural acoustic Kirchhoff-Boussinesq interactions with boundary feedback

We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.

Remarks on the energy inequality of a global $L^\infty$ solution to the compressible Euler equations for the isentropic nozzle flow

We study the compressible Euler equations in the isentropic nozzle flow. The global existence of an $L^{\infty}$ solution has been proved in (Tsuge in Nonlinear Anal. Real World Appl. 209: 217-238 (2017)) for large data and general nozzle. However, unfortunately, this solution does not possess finiteness of energy. Although the modified Godunov scheme is introduced in this paper, we cannot deduce the energy inequality for the approximate solutions. Therefore, our aim in the present paper is to derive the energy inequality for an $L^{\infty}$ solution. To do this, we introduce the modified Lax Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. We shall first deduce from the formula the energy inequality. Next, applying the compensated compactness method, the approximate solution converges to a weak solution. The energy inequality also holds for the solution as the limit. As a result, since our solutions are $L^{\infty}$, they possess finite energy and propagation, which are essential to physics.

Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D

In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function.

Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors

In this paper, we study the zero relaxation time limits to a one dimensional hydrodynamic model of two carrier types for semiconductors. First, we introduce the flux approximation coupled with the classical viscosity method to obtain the uniform $$L_{loc}^{p}, p \ge 1, $$ bound of the approximation solutions $$ \rho _{i}^{ \varepsilon ,\delta } $$ and other estimates of $$ (u_{i}^{ \varepsilon ,\delta }, E^{ \varepsilon ,\delta })$$ with the help of the high energy estimates (Jungel and Peng Comm Partial Differ Equ 58:1007–1033, 1999). Then, we apply the compensated compactness method coupled with the scaled variables technique (Marcati and Natalini Arch Ration Mech Anal 129:129–145, 1995) to prove the zero-relaxation-time limits with arbitrarily large initial data, and arbitrary adiabatic exponents $$ \gamma _{i} > 1$$ .

A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

Homogenization of some quasi-linear elliptic equations with gradient constraints

We prove a homogenization formula for quasi-linear elliptic equations with gradient constraints on a disperse set, within the framework of monotonic operator theory and compensated compactness methods.

Analysis of the Zero Relaxation Limit of Hyperbolic Balance Laws with Random Initial Data

We show the convergence of the zero relaxation limit of $2 \times 2$ systems of hyperbolic balance laws with stochastic initial data. Precisely, solutions converge to a solution of the local equilibrium approximation as the relaxation time tends to zero. The initial data are assumed to depend on finitely many random variables, and the convergence is then proved via the appropriate analogues of the compensated compactness methods used in treating the deterministic case. We also demonstrate the validity of this limit in the case of the semilinear $p$-system; the well-posedness of both the system and its equilibrium approximation is proved, and the convergence is shown with no a priori conditions on solutions. This model serves as a prototype for understanding how asymptotic approximations can be used as control variates for hyperbolic balance laws with uncertainty.

A note on the convergence of the solution of the Novikov equation

We consider the Novikov and Camass-Holm equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the \begin{document}$L^p$\end{document} setting.

Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions

We consider a coupled system of a linearly elastic body immersed in a flowing fluid which is modeled by means of the incompressible Navier–Stokes equations with mixed Dirichlet–Neumann-type boundary conditions. For this system we formulate an optimal control problem which amounts to a minimization under constraints of a hydro-elastic pressure at the interface between the two environments. The corresponding functional lacks convexity and radial unboundedness — a serious obstacle to the solution of the optimization problem. The approach taken to solve it is based on transforming the variable domain occupied by the fluid into a fixed domain corresponding to the undeformed elastic inclusion. This leads to a free boundary elliptic problem. Mathematical challenge also results from the fact that the corresponding quasilinear elliptic model is equipped with mixed (Zaremba type) boundary conditions, which intrinsically lead to compromised regularity of elliptic solutions. It is shown that under the assumption of small strains, the controlled structure is wellposed in suitable Sobolev’s spaces and the nonlinear control-to-state map is well defined and continuous. The obtained wellposedness result provides thus foundation for proving existence of an optimal control, where the latter is based on compensated compactness methods. The change in the boundary conditions makes the analysis different and substantially more challenging — particularly at the level of wellposedness of both uncontrolled and controlled dynamics. A key to the existence/uniqueness theory is a suitable localization of the spatial domain and of the resulting PDE. The geometry of the fluid domain plays a critical role in the arguments. The analysis of optimal control is nonstandard due to the lack of convexity and of radial unboundedness of the associated functional cost — main tools for proving weak lower-semicontinuity of the functional. This difficulty is dealt with the help of compensated compactness.

Well-posedness and Dispersive/Diffusive Limit of a Generalized Ostrovsky–Hunter Equation

We consider a generalized Ostrovsky–Hunter equation and the corresponding generalized Ostrovsky one with nonlinear dispersive effects. For the first equation, we study the well-posedness of entropy solutions for the Cauchy problem. For the second equation, we prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the first one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.

Global weak solutions for the chromatography system

In this paper, by using a new technique from the compensated compactness method, we study the Cauchy problem of the chromatography system of two equations, and obtain the existence of the global weak solutions when the regular BV estimate is assumed for only one characteristic field.

Existence of global [formula omitted] solutions to a system of conservation laws in chromatography with geometry effects

In this paper, we give the global existence of Lp bounded entropy solutions for the Cauchy problem of a system of conservation laws in chromatography with geometry effects. The main difficulty lies in establishing the Lp estimate of the viscosity solutions because the initial data may not be L∞ bounded and the principle of invariant regions developed by Conley–Chuey–Smoller is invalid here. We obtain the existence of the global weak solutions using the compensated compactness method and BV estimates on viscosity solutions.

Uniform boundedness in weak solutions to a specific dissipative system

We consider the L∞ weak solutions to a type of compressible Euler equation with dissipation effects. Several studies [6,14,30] have obtained the L∞ weak solutions to this type of system by using numerical schemes and the compensated compactness method. Therefore, the uniform boundedness of approximate solutions and the Hloc−1 compactness of the corresponding entropy dissipation measures must be considered. It should be noted that the obtained L∞ bounds typically increase over time. However, getting a time-independent uniform bound is important to consider the large time behavior of weak solutions. In this paper, by using invariant region theory, we prove that the L∞ weak solutions derived by the Lax–Friedrichs scheme are uniformly bounded in time.

Convergence of the regularized short pulse equation to the short pulse one

AbstractWe consider the regularized short‐pulse equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the short‐pulse one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the setting.

A note on convergence of the solutions of Benjamin–Bona–Mahony type equations

We consider the Benjamin–Bona–Mahony and the modified Benjamin–Bona–Mahony equations, which contains nonlinear dispersive effects. We prove that as the diffusion and dispersion parameters tend to zero, the solutions of these dispersive equations converge to the entropy ones of a scalar conservation law. The argument relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting.