# Guide Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems (Frontiers in Mathematics)

The results of this chapter are selected from . Global Existence of Spherically Symmetric Solutions 1. It follows from Lemma 1. Under the conditions of Theorem 1. We have the following representation: By the embedding theorem, 1. Under the conditions 1. Under conditions of Theorem 1. Proof of Theorem 1. With this structure 2. Consider the initial boundary value problem 2.

Introduction 35 Therefore we can translate the problem 2. The notation in the chapter is standard. Without danger of confusion, we will use the same symbol to denote the state functions as well as their values along the thermodynamic process, e. Our results read as follows, which are selected from [60, 64]. Global Existence and Exponential Stability 2.

We begin with the following lemma. Proof of Theorem 2. The proof is complete. Global Existence and Exponential Stability 1 Lemma 2. Thus we deduce from 2. Thus by the combination of 2. Global Existence and Exponential Stability which, together with 2. Global Existence and Exponential Stability By 2. Global Existence and Exponential Stability Inserting 2. Under the assumptions of Theorem 2.

Global Existence and Exponential Stability which, along with 2. Global Existence and Exponential Stability Lemma 2. Global Existence and Exponential Stability Multiplying 2. Global Existence and Exponential Stability Finally, using 2. Under assumptions of Theorem 2. We employ Theorems 2. Thus we infer from 2. Analogously, we get from 2. Global Existence and Exponential Stability Using 2. On the other hand, by 2. Bibliographic Comments 73 Proof. Among these cases we would like to mention two classes of models: For the former case, i. Global Existence and Exponential Stability The novelties of this chapter consist of the following aspects: The balance laws of mass, momentum, and energy in Lagrangian form are as follows: Regularity and Exponential Stability We consider a typical initial boundary value problem for 3.

The notation in this chapter is standard. Then from conservation of mass and boundary condition 3. Now our main results in this chapter read as follows.

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## Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems

Under assumptions of Theorem 3. Regularity and Exponential Stability 3. We begin with a technical lemma selected from . Only the estimate from above in 3. By use of 3. Therefore the desired estimate easily follows. Proof of Theorem 3. For convenience of the reader, we give a simple proof. Regularity and Exponential Stability Integrating 3. Regularity and Exponential Stability Since by 3. The balance of momentum 3. Regularity and Exponential Stability which, along with 3. Integrating the availability identity 3. Regularity and Exponential Stability Finally, multiplying 3.

Under the assumptions of Theorem 3. Then we derive from 3. The proof is similar to that of Lemma 2. Regularity and Exponential Stability Lemma 3. The proof is similar to that of Lemmas 2. The proofs of 3. We only need to prove 3. Regularity and Exponential Stability 4 Lemma 3. Bibliographic Comments 91 4 Lemma 3. For the case of an ideal gas, i. For the Cauchy problem with 3. Regularity and Exponential Stability [26, 27], Kawohl . Later on, Qin [49, 50, 51, 52, 54] established the same results as above on the global existence and exponential stability with some weaker assumptions.

From this point of view, 3. Thus the exponential stability of classical solutions is obtained, which is a new result for this model.

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All the above quantities are assumed to vary spatially only in the direction perpendicular to the plates. Global Existence and Exponential Stability rate, respectively. We consider a typical initial boundary value problem for 4. Then from conservation of mass and boundary condition 4. Hence we have improved the previous results in . Due to the involvement of a chemical reaction, the present situation is more complicated than that in Chapter 3, and more delicate and careful analyses are needed. Now we are in a position to state our main results which are chosen from .

Under the conditions in Theorem 4. In order to prove 4. So we obtain 4. Global Existence and Exponential Stability Multiply 4. Note the following simple consequence of 4. Thus, in view of 4. We will prove 4. The integrability of the mentioned functions is stated in 4. To deal with the remaining two derivatives, we note that by 4.

Note that with the boundary condition 4. Global Existence and Exponential Stability which, along with 4. Global Existence and Exponential Stability Lemma 4. Under the assumptions of Theorem 4. Combining this with 4. Since the proof is similar to that of Lemma 3. Global Existence and Exponential Stability Similarly to 4.

We derive from 4. Under assumptions of Theorem 4. The proof is similar to that of Lemma 3. Global Existence and Exponential Stability Integrating 4. Thus it follows from 4. This, along with Lemma 4. Global Existence and Exponential Stability 4. By virtue of 4.

Proof of Theorem 4. Thus, by Theorem 4. Thus, by Lemmas 4. Global Existence and Exponential Stability 4 Lemma 4. Similarly, we derive from 4. Using the embedding theorem, Lemmas 4. We infer from 4. When the material is a polytropic ideal linear viscous gas with constant viscosity and heat conductivity, e. Zheng and Qin  established the existence of maximal attractors for the problem 4. Global Existence and Exponential Stability The global existence and asymptotic behavior for the system 4.

However, Bebernes and Bressan  investigated the case that permits generation by a chemical reaction, so the system contains the chemical species equation 4.

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The results of this chapter are chosen from . The system under consideration in Lagrangian coordinates reads as follows: We consider system 5. However, under very high temperature and densities, equation 5. Without danger of confusion, we shall use the same symbol to denote the state functions as well as their values along a dynamic process, e. Introduction 4 Theorem 5.

Under assumptions of Theorem 5. Under the assumptions in Theorem 5. Thus we infer from 5. Proof of Theorem 4. Thus, by Theorem 4. Thus, by Lemmas 4. Global Existence and Exponential Stability 4 Lemma 4. Similarly, we derive from 4. Using the embedding theorem, Lemmas 4. We infer from 4. When the material is a polytropic ideal linear viscous gas with constant viscosity and heat conductivity, e.

Zheng and Qin  established the existence of maximal attractors for the problem 4. Global Existence and Exponential Stability The global existence and asymptotic behavior for the system 4. However, Bebernes and Bressan  investigated the case that permits generation by a chemical reaction, so the system contains the chemical species equation 4. The results of this chapter are chosen from .

The system under consideration in Lagrangian coordinates reads as follows: We consider system 5. However, under very high temperature and densities, equation 5. Without danger of confusion, we shall use the same symbol to denote the state functions as well as their values along a dynamic process, e. Introduction 4 Theorem 5. Under assumptions of Theorem 5. Under the assumptions in Theorem 5. Thus we infer from 5. Therefore, by Lemma 5. We readily obtain the next corollary. It follows from Corollary 5. We easily derive from 5.

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By using Lemmas 5. We derive from 5. Proof of Theorem 5. The proof is similar to that of Lemmas 3. The proof is similar to that of Lemma 4. We refer to Lemma 2. We refer to the proofs of Lemmas 4. We refer to the proof of Lemma 4. However, when radiation is Chapter 5. The whole problem under consideration when the matter is in local thermodynamical equilibrium LTE is thus a coupling between standard hydrodynamics for the matter and a radiative transfer equation for the photon distribution, through a suitable description, such as in plasma when the radiation is LTE with matter and velocities are not too large, a non-relativistic one temperature description is possible [44, 77].

For a viscous heat-conducting real gas with 5. In recent years, heat-conducting radiative viscous gas has drawn the attention of a number of mathematicians see, e. Among them, we would like to mention the work by Ducomet  with constitutive relations 5.

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However, we should note that there indeed exist some defects and even mistakes in the argument in , for example, in Lemmas , 13, etc. To prove Lemma 5. On the other hand, Theorem 5. Therefore it is necessary for us to investigate problem 5. The second is that in order to study the exponential stability, we need to establish uniform estimates depending only on the initial data, but independent of any length of time.

The main novelties of this chapter are as follows: Bressan, Global a priori estimates for a viscous reactive gas, Proc. A, , — Trivisa, Global solutions to the compressible NavierStokes equations with large discontinuous initial data, Comm. Sinica, 18 , 1—8.

Dafermos, Global smooth solutions to the initial boundary value problem for the equations for one-dimensional nonlinear thermoviscoelasticity, SIAM J. Hsiao, Global smooth thermomechnical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal. Triosia, On the motion of a viscous compressible radiative-reacting gas, Comm. Ducomet, Hydrodynamical models of gaseous stars, Reviews Math.

## Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems - PDF Free Download

Ducomet, A model of thermal dissipaption for a one-dimensional viscous reactive and radiative gas, Math. Feireisl, On the dynamics of gaseous stars, Arch. Feireisl, The equations of magnetohydrodynamics: Zlotnik, Stabilization for viscous compressible heatconducting media equations with nonmonotone state functions, C. I, ,— Zlotnik, Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state, J.

I, , — Zlotnik, Lyapunov functional method for 1D radiative and reactive viscous gas dynamics, Arch. Zhu, Asymptotic behaviour of the solution to the system for a viscous reactive gas, J. Luo, Large-time behavior of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. Edinburgh A, , — Jiang, On initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Jiang, On the asymoptotic behavior of the motion of a viscous, heatconducting, one-dimensional real gas, Math. Bibliography  S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm.

Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Flame, 50 , — Nishida, Global solutions to the initial boundary value problems for the equations of one-dimensional motion of viscous polytropic gases, J. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Kazhikhov, To a theory of boundary value problems for equations of one-dimensional nonstationary motion of viscous heat-conduction gases, in Boundary Value Problems for Hydrodynamical Equations, No. USSR, , 37—62 in Russian.

Kazhikhov, Cauchy problem for viscous gas equations, Siberian Math. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Luo, On the outer pressure problem of a viscous heat-conducting one dimensional real gas, Acta Math. Sinica, 13 , — A, 55 , — Nishida, The initial boundary value problems for the equations of motion of compressible viscous and heat-conductive gases, J. Weibel-Mihalas, Foundation of radiation hydrodynamics, Oxford Univ.

Press, New York, Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Qin, Global existence and asymptotic behavior of solutions to a nonlinear hyperbolic-parabolic coupled systems with arbitary initial data, Ph. Thesis, Fudan University, Qin, Global existence and asymptotic behavior of solutions to a system of equations for a nonlinear one-dimensional viscous, heat-conducting real gas, Chin.

Qin, Global existence and asymptotic behavior for the solutions to nonlinear viscous, heat-conductive, one-dimensional real gas, Adv. Qin, Global existence and asymptotic behavior of solution to the system in one-dimensional nonlinear thermoviscoelastcity, Quart. Qin, Exponential stability for a nonlinear one-dimensional heat-conductive viscous real gas, J. Qin, Global existence for the compressible Navier Stokes equations with Dirichlet boundary conditions, J. Qin, Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity, IMA J.

Bibliography  Y. Models and Methods Appl. Huang, Global existence and exponential stability for the ph power viscous reactive gas, Nonlinear Anal. Qin and Jaime E. Elasticity, 66 , — Elasticity, 5 , — Wen, Global existence of spherically symmetric solutions for nonlinear compressible Navier-Stokes equations, J. Liu, On the Cauchy problem for one-dimensional compressible Navier-stokes equations, Portugaliae Mathematica, 64 , 87— Secchi, On the motion of gaseous stars in the presence of radiation, Comm.

Tani, Global solutions to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas, J. Edinburgh, A , — Yang, Local existence with physical vacuum boundary condition to Euler equations with damping, J. Yanagi, Existence of periodic solutions for a one-dimensional isentropic model system of compressible viscous gas, Nonlinear Anal. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart.

Shen, Global solutions to the Cauchy problem of the equations of one-dimensional thermoviscoelasticity, J. Zimmer, Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy, J. Global controllability and stabilization of nonlinear systems. Strongly Coupled Coulomb Systems. Strongly coupled coulomb systems. Semiclassical states for weakly coupled nonlinear Schrodinger systems. Integrability of Nonlinear Systems. Foundations of global nonlinear analysis. Approximation of nonlinear evolution systems. Control of sandwich nonlinear systems.

Nonlinear Dynamics of Production Systems. Stability of nonlinear control systems. Identification of nonlinear physiological systems. Linear models of nonlinear systems. Stability of Nonlinear Control Systems.

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