Ya-Guang Wang

Well-posedness of the Prandtl equation in Sobolev spaces

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

Quasi-neutral limit of the non-isentropic Euler-Poisson system

This paper is concerned with multi-dimensional non-isentropic Euler–Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified.

Nonlinear Well-Posedness and Rates of Decay in Thermoelasticity with Second Sound

The Cauchy problem in nonlinear thermoelasticity with second sound in one space dimension is considered. Due to Cattaneos law, replacing Fouriers law for heat conduction, the system is hyperbolic. The local well-posedness as a strictly hyperbolic system is investigated first, and then the relation between energy estimates for non-symmetric hyperbolic systems and well-posedness are discussed. For the global small solution, the long time behavior is described and the decay rates of the L2-norm are obtained.

Microlocal analysis in nonlinear thermoelasticity

Abstract This paper is devoted to the study of the propagation of singularities for semilinear hyperbolic–parabolic coupled systems in thermoelasticity. First, a linear transformation involving pseudodifferential operators for unknowns is introduced to decouple the hyperbolic and the parabolic operators. By using the decoupled system, we obtain that the semilinear system of thermoelasticity has finite speeds for the propagation of singularities. Furthermore, it is proved that the microlocal singularities of solutions to this semilinear hyperbolic–parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of nonsmooth pseudodifferential operators. Finally, the Cauchy problem for the semilinear system of thermoelasticity is investigated when the initial data have singularities.

Fractal Dimension of Attractors for a Stochastic Wave Equation with Nonlinear Damping and White Noise

Abstract We investigate the existence of compact random attractors and their fractal dimension for the random dynamical system associated with a stochastic wave equation with nonlinear damping and white noise.

Zero-Viscosity Limit of the Linearized Compressible Navier-Stokes Equations with Highly Oscillatory Forces in the Half-Plane

We study the asymptotic behavior of the solution to the linearized compressible Navier--Stokes equations with highly oscillatory forces in the half-plane with nonslip boundary conditions for small viscosity. The wavelength of oscillation is assumed to be proportional to the square root of the viscosity. By means of asymptotic analysis, we deduce that the leading profiles of the solution have four terms: the first one is the outflow satisfying the linearized Euler equations, the second one is an oscillatory wave propagated along the characteristic field tangential to the boundary associated with the linearized Euler operator in the half-plane, the third one is a boundary layer satisfying a linearized Prandtl equation, the fourth one represents the oscillation propagated in the boundary layer, and it is described by an initial-boundary value problem for an Poisson--Prandtl coupled system. By using the energy method and mode analysis, we obtain the well-posedness of this Poisson--Prandtl coupled problem, and...

Boundary layers and quasi-neutral limit in steady state Euler–Poisson equations for potential flows

We study the quasi-neutral limit in the steady state Euler–Poisson system for potential flows. Boundary layers occur when the boundary conditions are not in equilibrium. We perform a formal asymptotic expansion of solutions and derive the boundary layer equations. Under the subsonic condition on the boundary and the smallness assumption on the data, the existence, uniqueness and exponential decay of the boundary layer profiles are proved by applying the centre manifold theorem to a dynamical system. We also give a rigorous justification of the asymptotic expansion up to first order in one space dimension.

On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308, 2014) on the well-posedness theory for the three-dimensional Prandtl equations with a special structure.

decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

L p - L q decay estimates of solutions to the Cauchy problem of linear thermoelastic systems with second sound in one space variable will be studied in this paper. First, by dividing the frequency of phase space of the Fourier transformation into different regions, the asymptotic behavior of characteristic roots of the coefficient matrix is obtained by carefully analyzing the effect of the different regions. Second, with the help of the information on the characteristic roots and by using the interpolation theorem, the L p - L q decay estimate of solutions to the Cauchy problem of the linear thermoelastic system with second sound in one space variable is obtained.

Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws

We are concerned with entropy solutions of hyperbolic systems of conservation laws in several space variables. The Euler equations of gas dynamics and magnetohydrodynamics (MHD) are prototypes of hyperbolic conservation laws. In general, there are two types of discontinuities in the entropy solutions: shock waves and characteristic discontinuities, in which characteristic discontinuities can be either vortex sheets or entropy waves. In gas dynamics and MHD, across a vortex sheet, the tangential velocity field has a jump while the normal velocity is continuous; across an entropy wave, the entropy has a jump while the velocity field is continuous. A vortex sheet or entropy wave front is a part of the unknowns, which is a free boundary. Compressible vortex sheets and entropy waves, along with shock and rarefaction waves, occur ubiquitously in nature and are fundamental waves in the entropy solutions to multidimensional hyperbolic conservation laws. The local stability of shock and rarefaction waves has been relatively better understood. In this paper we discuss the stability issues for vortex sheets/entropy waves and present some recent developments and further open problems in this direction. First we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics and some recent developments for a rigorous mathematical theory on their nonlinear stability/instability. Then we review our recent study and present a supplement to the proof on the nonlinear stability of compressible vortex sheets under the magnetic effect in three-dimensional MHD. The compressible vortex sheets in three dimensions are unstable in the regime of pure gas dynamics. Our main concern is whether such vortex sheets can be nonlinearly stabilized under the magnetic fields. To achieve this, we first set up the current-vortex sheet problem as a free boundary problem; then we establish high-order energy estimates of the solutions to the linearized problem, which shows that the current-vortex sheets are linearly stable when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields; and finally we develop a suitable iteration scheme of the Nash–Moser–Hormander type to obtain the existence and nonlinear stability of compressible current-vortex sheets, locally in time. Some further open problems and several related remarks are also presented.