Nonlinear and Variational Analysis and their ApplicationsView this Special Issue
Global Well Posedness for the Thermally Radiative Magnetohydrodynamic Equations in 3D
In this paper, we study the thermally radiative magnetohydrodynamic equations in 3D, which describe the dynamical behaviors of magnetized fluids that have nonignorable energy and momentum exchange with radiation under the nonlocal thermal equilibrium case. By using exquisite energy estimate, global existence and uniqueness of classical solutions to Cauchy problem in or are established when initial data is a small perturbation of some given equilibrium. We can further prove that the rates of convergence of solution toward the equilibrium state are algebraic in and exponential in under some additional conditions on initial data. The proof is based on the Fourier multiplier technique.
In the study of plasma physics, due to the high temperature and high pressure environment, the motion of charged particles flow is usually regarded as compressible fluids, and their dynamics is very often shaped and controlled by magnetic fields and high temperature radiation effects. Meanwhile, it is known that the radiation energy is carried by photons. When the distribution of photon is almost isotropic, based on the standard hydrodynamics, such dynamics can be described by the following 3D thermally radiative magnetohydrodynamic equations (cf. [1, 2]): where , , , , for denote the mass density, velocity field of the fluid, mass temperature, magnetic field, and radiation field, respectively, and denotes the material pressure. The spatial domain or . The parameter is the perfect gas constant, is the specific heat at constant volume, and is the heat conductivity coefficient; and are the viscosity coefficients of the flow satisfying and and is the magnetic diffusion coefficient. Throughout this paper, we assume that are all positive constants. is the deformation tensor
When the magnetic is ignored (i.e., in (1)), system (1) can be reduced to the nonequilibrium diffusion approximation model in radiation hydrodynamics. This model describes the energy flow due to radiative process in a semiquantitative sense and is particularly accurate if the specific intensity of radiation is almost isotropic (cf. [3–5]). There are some mathematical results on this model. For the global existence of smooth solution for one-dimensional case, see ; for the global well-posedness and large time behavior of classical solutions for multidimensional case, see . For the inviscid case, in , the authors considered a 1D model and showed the existence of shock profiles for inviscid nonequilibrium gases provided that the initial strength is suitably small. For the local existence of smooth solutions for multidimensional system, see . System ((1)) is the compressible MHD equations coupled with the radiative transport equation with nonlocal terms and is very difficult to solve both numerically and analytically. Ducomet and Feireisl consider the thermally radiative MHD system first and show the existence of global weak solutions for the multidimensional case in  (also see, for instance, Li and Guo ). For the one-dimensional case, this model has been studied by many authors under the various growth constraints on the heat conductivity [12–14].
In this paper, we are focused on the asymptotic and global existence of classical solutions of system (1) with the initial data:
It is easy to check that is an equilibrium state of (1). Therefore, it is natural to introduce the transforms
Without loss of generality, we assume the positive constants . Then we can rewrite the system (1) as with initial data
Then, the main results in this paper read as follows:
Theorem 2. Under the conditions of Theorem 1, if we further assume that is sufficiently small, then for all .
The proof of the global existence of classical solution to (5)–(10) relies on the global a priori estimates together with the local existence of classical solutions and continuum argument. The main difficulty in establishing prior estimates in high-order Sobolev spaces is how to control the linear term in (5)–(9), such as , in (8) and in (1.6). We develop the method in  and use the structure of system (5)–(9) itself to construct novel dissipation term to overcome this difficulty. To prove Theorem 2, we first use a Fourier multiplier technique to establish the time-decay property of linearized system (47)–(52). Then, the time decay rate can be given by combining the global a priori estimate obtained in Theorem 1 and the above property and applying the energy estimate technique to the nonlinear problem (5)–(10), whose solutions can be represented by the solution-semigroup operator for the linearized system (47)–(52) by using the Duhamel principle. Here, some nonlinear terms of magnetic field involved in (6) and (8) may lead to difficulties to gain the desired rate of convergence of solutions. Thus, we will construct some novel functionals such as (79) and (80) and adopt with modification some techniques motivated by [16–18] combined to some vector analysis formula to obtain expected decay rates.
The remainder of this paper is organized as follows. In Section 2, we derive the uniform-in-time a priori estimates and then establish the existence of a global classical solution. In Section 3, we investigate the decay rates of solutions. In Section 4, we adapt our proof to the periodic domain case. Throughout this paper denotes a positive (generally large) constant and a positive (generally small) constant, where both and may take different values in different places. The symbol means for a generic constant . For simplicity, we shall use to denote norm .
2. Global Existence
2.1. A Priori Estimates
Now, we begin to establish the global a priori estimates in the case of the whole space under the assumption where is a generic constant small enough and is the smooth solution to the Cauchy problem (5)–(10) on for . Firstly, we list two important lemmas in Sobolev space.
Lemma 5 (Moser-type calculus inequalities) (see ). Let be an integer. Suppose , and . Then, for all multi-index with , we have Then, we begin to give the priori estimate of .
Proof. Multiplying (5)–(9) by , and and then taking integration and summation, we get For to , using Hölder’s and Sobolev’s inequalities, we have For , , under the assumption (16), one also has At last, for and , we have Plugging all the above estimates into (20), we obtain (19).
Proof. Applying with to (5)–(9) and multiplying by , and , respectively, then taking integration and summation, we have where denotes the commutator for two operators and ; is constant depending only on and . We now bound each term on the right-hand side of (25). Utilizing Lemma 5, we get For , we have with a small constant, where the first inequality follows that for , and Sobolev’s and Young’s inequalities were further used. Similarly, we have For , we have By Lemma 4, we obtain since Similarly, we can deduce that Therefore, Using Hölder’s, Sobolev’s, and Young’s inequalities and Lemma 4, we can get following bounds:
For the remaining terms, under the assumption (16), one also has
Next, we will give the dissipation rate of .
2.2. Proof of Global Existence
In this section, we will show there exists a unique global in time solution to the problem (5)–(10). Firstly, combining estimates obtained in Lemmas 6–8, one can finish the proof of uniform-in-time a priori estimates as follows. Define a total temporal energy functional and the corresponding dissipation rate functional by where is a small constant. Under the assumption (16), then holds true uniformly for all . Furthermore, summing ((19)) and ((24)) and × ((37)) and noticing thatis sufficiently small, we have for all . With the help of (16), one has with being small enough. Then, the time integration of (44) yields for all . Besides, (16) can be justified by choosing which is sufficiently small. For brevity, the proof for local existence of smooth solutions is omitted. Then, the global existence and uniqueness of solutions follow from (45) together with the local existence as well as application of the continuity argument.
3. Convergence Rates
In this section, we consider the convergence rate of solutions obtained in Theorem 1. In order to obtain the desired decay rate estimates in Theorem 2, we firstly consider the linearized system corresponding to (5)–(9): with initial data
Denote by to be the solution of the Cauchy problem (47)–(52), then can be presented as where is named as the solution operator of (47)–(52) and . Then, we utilize the energy method to Cauchy problem (47)–(52) in the Fourier space to present that there is a time-frequency Lyapunov functional which is equivalent to . This estimate can help us to establish the time decay property of as follows.
Theorem 9. Let. For any with and , hold for all .
Proof. By taking Fourier transforming in for (47)–(51), one has
where , is the imaginary unit.
Multiplying (55)–(59) by , , respectively, its real part gives Multiplying (56) by , utilizing integration by parts in , and replacing by (55), one has here means the complex inner product. For the real part of (61) and with the help of Cauchy-Schwarz inequality, one has with being a small constant. Multiplying it by , we conclude that there exists such that Now, we define the time-frequency Lyapunov functional as where is sufficiently small. It also holds that . Moreover, by suitably choosing constants , the sum of equations (19), (24), × (37) gives the linear combination (60) + × (63) which gives (60)
Lemma 10. Given any and , for all .