Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the 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 weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Lp-Lq time decay estimate to the solution of the Cauchy problem of the system of equations for nonlocal model of hyperbolic thermoelasticity theory

The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e(x), and heat flux q(x), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. Using the modified Cagniard–de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three–dimensional space. Basing on the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the L^p−L^q time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lp – Lq time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body

The aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard-de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three-dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the L^p–L^q time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L¹ to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L¹-error bound in is established.     

The Cauchy problem for multiphase first-contact miscible models with viscous fingering

In this paper, the Cauchy problem for multiphase first-contact miscible models with viscous fingering, is studied and a global weak solution is obtained by using a new technique from the Div–Curl lemma in the compensated compactness theorem. This work extends the previous works, (Juanes and Blunt, 2006; Barkve, 1989), which provided the analytical solutions and the entropy solutions respectively, of the Riemann problem, and (Lu, 2013), which provided the global solution of the Cauchy problem for the Keyfitz–Kranzer system.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

A class of large solution of 2D tropical climate model without thermal diffusion

In this paper, we consider the Cauchy problem of 2D tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data whose L^∞ norm can be arbitrarily large.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

Convergence rates to nonlinear diffusion waves for weak entropy solutions to<Emphasis Type="Italic">p</Emphasis>-system with damping

This paper studies the asymptotic behavior of weak entropy solutions to the Cauchy problem of the so-called p-system with damping. The convergence rates to nonlinear diffusion waves for weak entropy solutions are obtained in L∞norm or L² -norm. These convergence rates are the same to the decay rates of smooth solution obtained by Nishihara. They are proved by using the vanishing viscosity method and the elementary L²-energy method.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

The global weak solutions to the Cauchy problem of the generalized Novikov equation

In this paper, we study the Cauchy problem of the generalized Novikov equation. We first show that under suitable condition, the strong solution exists globally via some a priori estimates. Then, we prove the existence and uniqueness of global weak solutions by the approximation method. We also obtain the exact peaked solutions.     

The Cauchy Problem for an Axially Symmetric Equation and the Schwarz Potential Conjecture for the Torus

We present our results in this paper in two parts. In the first part, we consider the Cauchy problem for the axially symmetric equation with entire Cauchy data given on an initial plane (see Eq. (2.1)). We solve the Cauchy problem and obtain its solutions in two cases, depending on whether k is a positive even integer or k is a positive odd integer. For k odd, we demonstrate that the solution has more singularities due to the propagation of the singularities of the coefficients. In the second part, the Cauchy problem for the same equation is considered, but instead, its entire Cauchy data are given on an initial sphere (see Eq. (3.1)). Whenever k is a positive even integer, we obtain the global existence of the solution and determine all possible singularities. Whenever k is a positive odd integer, we discuss both local and global solutions. As a consequence of our results in this paper, we show that the Schwarz Potential Conjecture (see the Introduction) for the even dimensional torus is true.     

Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type

In this paper, we study the global L^∞ solutions for the Cauchy problem of nonsymmetric system (1.1) of Keyfitz-Kranzer type. When n=1, (1.1) is the Aw-Rascle traffic flow model. First, we introduce a new flux approximation to obtain a lower bound ρε,δ?δ>0 for the parabolic system generated by adding “artificial viscosity” to the Aw-Rascle system. Then using the compensated compactness method with the help of L¹ estimate of wε,δx(⋅,t) we prove the pointwise convergence of the viscosity solutions under the general conditions on the function P(ρ), which includes prototype function , where γ∈(−1,0)∪(0,∞), A is a constant. Second, by means of BV estimates on the Riemann invariants and the compensated compactness method, we prove the global existence of bounded entropy weak solutions for the Cauchy problem of general nonsymmetric systems (1.1).     

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L ¹-estimate between the entropy solution and the geometric optics expansion function is bounded by O(?²), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.     

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.     

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier-Stokes equations with damping α|u|^β−1u(α>0) has global weak solutions for any β?1, global strong solution for any β?7/2 and that the strong solution is unique for any 7/2?β?5.     

Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type

In this paper, we use the compensated compactness method with BV estimates on the Riemann invariants to obtain the global existence of bounded entropy weak solutions for the Cauchy problem of a symmetric system of Keyfitz-Kranzer type.     

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.