Global Weak Entropy Solution Research Articles

Global Weak Entropy Solution of Nonlinear Ideal Reaction Chromatography System and Applications

Global Weak Entropy Solution of Nonlinear Ideal Reaction Chromatography System and Applications

On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

In this paper, we consider global weak solutions to compressible Navier–Stokes–Korteweg equations with density dependent viscosities, in a periodic domain $$\Omega = \mathbb T^3$$ , with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition, introduced in D. Bresch, A. Vasseur and C. Yu (Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities. arXiv:1905.02701 , 2019). Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier–Stokes system with a drag term.

Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

Global weak entropy solutions to the Euler–Poisson system with spherical symmetry

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform [Formula: see text] estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

Asymptotic Behavior of Global Entropy Solutions for Nonstrictly Hyperbolic Systems with Linear Damping

We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that ast→∞the entropy solutions tend to zero in theLpnorm.

Global weak entropy solution to Doi–Saintillan–Shelley model for active and passive rod-like and ellipsoidal particle suspensions

We prove the existence of the global weak entropy solution to the Doi–Saintillan–Shelley model for active and passive rod-like particle suspensions, which couples a Fokker–Planck equation with the incompressible Navier–Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd−1)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd−1) weak solution in two and three dimensions and the uniqueness in two dimension.

Global solutions to one-dimensional shallow water magnetohydrodynamic equations

In this paper, we study the Cauchy problem for the one-dimensional shallow water magnetohydrodynamic equations. The main difficulty is the case of zero depth (h=0) since the nonlinear flux function P(h) is singular and the definition of solution is not clear near h=0. First, assuming that h has a positive and lower bound, we establish the pointwise convergence of the viscosity solutions by using the div–curl lemma from the compensated compactness theory to special pairs of functions (c,fε), and obtain a global weak entropy solution. Second, under some technical conditions on the initial data such that the Riemann invariants (w,z) are monotonic and increasing, we introduce a “variant” of the vanishing artificial viscosity to select a weak solution. Finally, we extend the results to two special cases, where P(h) is for the polytropic gas or for the Chaplygin gas.

Existence of Global Weak Entropy Solutions to Some Nonstrictly Hyperbolic Systems

In this paper, the Cauchy problem for nonstrictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery is studied, and a short proof of global existence of weak entropy solutions is obtained by using a new technique from the compensated compactness theorem coupled with the total variation estimate on one family of Riemann invariants. This work extends in some sense the previous works [T. Barkve, SIAM J. Appl. Math., 49 (1989), pp. 789--798; T. Johansen and R. Winther, SIAM J. Math. Anal., 19 (1988), pp. 541--566] about the global existence of weak solutions for the Riemann problem and by the author [J. Funct. Anal., 261 (2011), pp. 2797--2815] about the global weak solution of the Cauchy problem for an extended hyperbolic system of Keyfitz--Kranzer or Aw--Rascle type.

Global weak solutions to the non-relativistic radiation hydrodynamical equations with isothermal fluids

We prove the existence of global entropy solutions in L∞ to the one-dimensional Euler–Boltzmann equations for compressible isothermal fluids and multidimensional Euler–Boltzmann equations for compressible isothermal fluids with spherical symmetry in radiation hydrodynamics. The global weak entropy solutions are constructed using the shock capturing scheme of Godunov type with cut-off technique. The global existence of weak entropy solutions in L∞ with arbitrarily large initial data even containing the vacuum is established with the aid of the compensated compactness method.

GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM

In this paper we consider the relativistic waterbag continuum which is a useful PDE for collisionless kinetic plasma modeling recently developed in Ref. 11. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they are very useful for analytical theory and numerical simulations of laser-plasma and gyrokinetic physics.10–16, 56, 57The relativistic waterbag continuum is derived from two phase-space variable reductions of the relativistic Vlasov–Maxwell equations through the existence of two underlying exact invariants, one coming from physics properties of the dynamics is the canonical transverse momentum, and the second, named the "water-bag" and coming from geometric property of the phase-space is just the direct consequence of the Liouville Theorem. In this paper we prove the existence and uniqueness of global weak entropy solutions of the relativistic waterbag continuum. Existence is based on vanishing viscosity method and bounded variations (BV) estimates to get compactness while proof of uniqueness relies on kinetic formulation of the relativistic waterbag continuum and the associated kinetic entropy defect measure.

Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics

The Cauchy problem for the one-dimensional Euler-Boltzmann equations in radiation hydrodynamics is studied. The global weak entropy solutions are constructed using the Godunov finite difference scheme. The global existence of weak entropy solutions in L ∞ L^\infty with arbitrarily large initial data is established with the aid of the compensated compactness method.

Weak solutions to isothermal hydrodynamic model for semiconductor devices

In this paper, the Cauchy problem of the isothermal hydrodynamic model for semiconductor devices is investigated. The existence of global weak entropy solutions with large initial data is obtained by using a modified fractional step Lax–Friedrichs scheme and the theory of compensated compactness. As a byproduct, the existence of entropy solutions to the Cauchy problem of the isentropic hydrodynamic model for a semiconductor with infinite mass is also proved.

GLOBAL SOLUTION FOR ROTATING COMPRESSIBLE PLASMA FLOW WITH LARGE DATA

The initial boundary value problem for the Euler–Poisson equations of the two-dimensional compressible rotating plasma flow with large data in L∞is studied in the isothermal case. The shock capturing method is used to construct the approximate solution. The uniform estimate and the H-1estimate of the entropy dissipation measures are obtained, and the compensated compactness method is applied to show the convergence of the approximate solution. The nonlocal effect of the Poisson equation is analyzed. The limit of the approximate solution is a weak entropy solution. Therefore the global weak entropy solution in L∞to the Euler–Poisson equations for rotating plasma flow is constructed and the global existence is established.

Large BV solutions to the compressible isothermal Euler–Poisson equations with spherical symmetry

The Euler–Poisson equations for isothermal flows in charge transport are studied. The global weak entropy solution with spherical symmetry is constructed for arbitrarily large initial data in BV. The global solution is in BV and has large total variations. One difficulty is that the system of Euler–Poisson equations is a hyperbolic–elliptic coupled system, and the property of finite propagation speed does not hold. A modified Glimm scheme is used to construct the approximate solution of the Euler equations while the Poisson equation is approximated by an integral of the solution to the Euler equations. Other difficulties include the geometrical structure, the damping and some additional source terms, which make the analysis more complicated and difficult. To overcome these difficulties, the solution of a discrete ordinary differential equation is used to construct the approximate solution. It is proved that vacuum will not occur in the solution, which is important to obtain the compactness of the approximate solution in BV.

𝐿¹-convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary

This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws u t + f ( u ) x = 0 {u_t} + f{\left ( u \right )_x} = 0 on [ 0 , ∞ ) × [ 0 , ∞ ) \left [ {0, \infty } \right ) \times \left [ {0, \infty } \right ) , with the initial data u ( x , 0 ) = u 0 ( x ) u\left ( {x, 0} \right ) = {u_0}\left ( x \right ) and the boundary data u ( 0 , t ) = u − u\left ( {0, t} \right ) = {u_ - } , where u − {u_ - } is a constant, u 0 ( x ) {u_0}\left ( x \right ) is a step function with a discontinuous point, and f ∈ C 2 f \in {C^2} satisfies f > 0 , f ( 0 ) = f ′ ( 0 ) = 0 f > 0, f\left ( 0 \right ) = f’\left ( 0 \right ) = 0 . The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary x = 0 x = 0 and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by O ( ϵ 1 / 2 + ϵ | I n ϵ | + ϵ ) O\left ( {\epsilon ^{1/2}} + \epsilon \left | {In\epsilon } \right | + \epsilon \right ) in L 1 {L^1} -norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is O ( ϵ | I n ϵ | + ϵ ) O\left ( \epsilon \left | {In\epsilon } \right | + \epsilon \right ) . In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to O ( ϵ ) O\left ( \epsilon \right ) . The proof is given by a matching method and the traveling wave solutions.

Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and Sine-Gordon type

Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and Sine-Gordon type

Global solutions to the Euler-Poisson equations of two-carrier types in one dimension

The global solution to the one-dimensional Euler-Poisson equations of two-carrier types is constructed by the fractional Godunov method. The compensated compactness framework is applied to prove the convergence of the approximate solutions. Some sharp form of the uniform estimate is made to overcome the difficulties due to the coupling terms. The existence of the global weak entropy solution with arbitrarily large initial data is established.

The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.