We consider a general system of hyperbolic-parabolic balance laws, which contains both second-order dissipation represented by a viscosity matrix and lower-order damping or relaxation. As dictated by physics, the viscosity matrix and the Jacobian matrix of the lower-order term are usually rank deficient. We study the Green’s function of its linearization around a constant equilibrium state. The Green’s function is constructed using heat kernels along equilibrium characteristic directions and delta functions along other predetermined directions. Detailed error estimates are given in the space-time pointwise sense. Our main result is then applied to two different models of polyatomic gas flows. The first one is a model of translational and vibrational non-equilibrium flows with a general setting in the fundamental thermodynamic relations. The second one is a two-temperature Navier–Stokes type model for polyatomic gas flows, proposed by Aoki et al., with specific equations of state. For each model, we give an explicit formulation of Green’s function for the Cauchy problem.

We study long time behavior of polyatomic gas flows in both translational and vibrational non-equilibrium. The author previously established global existence of solution and obtained optimal L-2 time-decay rates for the solution towards an equilibrium state for the Cauchy problem. The current paper is a continuation in studying the solution behavior. An asymptotic solution is constructed explicitly using a heat kernel along the particle path and two Burgers kernels along the equilibrium acoustic directions. Convergence of the exact solution to the asymptotic solution is studied in a pointwise sense in both space and time to give a complete picture of wave propagation. The study lays a foundation for a future work on solution behavior around a shock wave, a mechanism that induces Richtmyer-Meshkov instability in mixing problems .

This paper considers the global dynamics of classical solutions to an initial-boundary value problem of the system of viscous balance laws arising from chemotaxis in one space dimension: u(t) - (uv)(x) = u( xx) + u(1 - u), x is an element of (a, b), t >0, v( t) -(u + v (2) )(x) = v(xx), x is an element of (a, b), t > 0 . The system of equations is supplemented with time-dependent influx boundary condition for u and homogeneous Dirichlet boundary condition for v . Under suitable assumptions on the dynamic boundary data, it is shown that classical solutions with generic initial data exist globally in time. Moreover, the solutions are shown to converge to the constant equilibrium ( 1 , 0), as t -> infinity . There is no smallness assumption on the initial data. This is the first rigorous mathematical study of the model subject to dynamic Neumann boundary condition, and generalizes previous works in content and technicality.

The paper is a continuation of the author’s work in [14]. We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We consider Cauchy problems with Cauchy data not bounded away from the logarithmic singularity. The model can be converted into a 2× 2 system of hyperbolic-parabolic balance laws by inverse Hopf-Cole transformation, with Cauchy data connecting two different end states. The converted form was studied in [14] when Cauchy data are near a diffusive contact wave. The current paper is to study the scenario under the original model to gain understanding of the evolution of physical quantities when the logarithmic singularity plays an intrinsic role. For all three cases, singularity at -∞ , at +∞ , and at ±∞ , we obtain a clear picture of time asymptotic behavior of solutions.

We study BV solutions for a 2×2 system of hyperbolic balance laws. We show that when initial data have small total variation on (−∞,∞) and small amplitude, and decay sufficiently fast to a constant equilibrium state as |x|→∞, a Cauchy problem (with generic data) has a unique admissible BV solution defined globally in time. Here the solution is admissible in the sense that its shock waves satisfy the Lax entropy condition. We also study asymptotic behavior of solutions. In particular, we obtain a time decay rate for the total variation of the solution, and a convergence rate of the solution to its time asymptotic solution. Our system is a modification of a Keller-Segel type chemotaxis model. Its flux function possesses new features when comparing to the well-known model of Euler equations with damping. This may help to shed light on how to extend the study to a general system of hyperbolic balance laws in the future.

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.

We consider a 2×2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x→±∞, reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L2 and L∞ convergence rates are obtained. We first convert the model into a 2×2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.

We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. It is a 2×2 reaction-diffusion system describing the interaction of cells and a chemical signal. We study Cauchy problem for the original system and its transformed system, which is one of hyperbolic-parabolic balance laws. Our initial data are generic perturbations of a constant ground state, i.e. the initial mass of perturbation is non-zero. In the case of non-diffusive chemical, we obtain optimal L2 time decay rates for the solution with finite initial data. In the case of diffusive chemical, optimal L2 rates are also obtained with additional assumption on the smallness of the initial amplitude but still allowing large oscillation.

We consider a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. It is a 2 by 2 system describing the interaction of cells and a chemical signal. We study Cauchy problem with finite initial data, i.e., without the commonly used smallness assumption on initial perturbations around a constant ground state. We survey a sequence of recent results by the authors on the existence of global-in-time solution, long-time behavior, vanishing coefficient limit and optimal time decay rates of the solution.

We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal [Formula: see text] decay rates for [Formula: see text]. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively.

We study time asymptotic behavior of solutions for a general system of hyperbolic-parabolic balance laws in m space dimensions, m >= 2. The system has physical viscosity matrices. Besides, there is a lower order term to account for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. We study Cauchy problem around a constant equilibrium state. Under a set of reasonable assumptions, existence of solution global in time has been established recently, and L-p decay rates (p >= 2) of the solution to the constant equilibrium state have been obtained. In this paper we further study the large time behavior of the solution. We show that it is time-asymptotically approximated by the solution of the corresponding linear system with the same initial data. For p >= 2, optimal L-p convergence rates to the asymptotic solution are obtained. These rates are faster by (t+1)(-1/2) (or (t+ 1)(-1/2) ln(t+ 2) if m = 2) when comparing to the convergence rates to the constant equilibrium state. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result is new even for the special case of hyperbolic balance laws.

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibrium state, and obtain a detailed, pointwise description of the Green’s function. The result provides a complete solution picture for the linear problem. It also helps to shed light on small solutions of the nonlinear system.

We study a general system of hyperbolic-parabolic balance laws in [Formula: see text] space dimensions ([Formula: see text]). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.

We study the equations describing the motion of a thermal non-equilibrium gas in three space dimensions. It is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that the Shizuta-Kawashima criterion is violated. This implies that a perturbation of a constant equilibrium state consists of two parts: one decays in time while the other stays. In fact, the entropy wave grows weakly along the particle path as the process is irreversible. We study thermal properties related to the well-posedness of the nonlinear system. We also obtain a detailed pointwise estimate on the Green’s function for the Cauchy problem when the system is linearized around an equilibrium constant state. The Green’s function provides a complete picture of the wave pattern, with an exact and explicit leading term. Comparing with existing results for one dimensional flows, our results reveal a new feature of three dimensional flows: not only does the entropy wave not decay, but the velocity also contains a non-decaying part, strongly coupled with its decaying one. The new feature is supported by the second order approximation via the Chapman-Enskog expansions, which are the Navier-Stokes equations with vanished shear viscosity and heat conductivity.

We study the equations describing the motion of a thermal non-equilibrium gas with one non-equilibriummode. In three space dimensions it is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that Shizuta-Kawashima criterion is violated. However, there is a significant difference between one dimensional and three dimensional flows in how the criterion is violated. As a consequence, the velocity components in their solutionsbehave differentlywhile thermal dynamic variables share common properties.

In this paper we are interested in a general class of hyperbolic balance laws. Within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the Kawashima–Shizuta condition, we study the large time behavior of the solution to the Cauchy problem in one space dimension. For a solution around a constant equilibrium state, we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions. We estimate the remainder in the pointwise sense both in space and in time. The decay rates in various directions are optimal, which further give the optimal Lp rates with 1≤p≤∞. Applications to several examples including the Kerr–Debye model are given.

We consider a general system of hyperbolic balance laws in m space dimensions (m >= 1). Under a set of conditions we establish the existence of global solutions for the Cauchy problem when initial data are small perturbations of a constant equilibrium state. The proposed assumptions in this paper are different from those in literature for the system. Instead, our assumptions are parallel to those used in the study of hyperbolic parabolic systems. In one space dimension our assumptions are natural extensions of those used in the study of the Green's function of the linearized system. They are also sufficient to the study of large time behavior in the pointwise sense for the nonlinear system, carried out in a different paper.