Sobolev regularity solutions for a class of singular quasilinear ODEs

1 Introduction and main results

In this paper, we consider the following singular second-order ordinary differential equations

with the initial-boundary condition

and the constraint condition

where positive constants 0 < ε < 1 and μ, λ, C > 0, f(r) denotes an external force. Obviously, there is a singular coefficient 2r in equation (1.1), and it is a class of dissipative-type ODEs with the dissipative term 2r u.

We state the main result of this paper.

Above singular quasilinear second-order ordinary differential equations with the constraint condition comes from the the three dimensional Navier-Stokes system for compressible fluids in the steady isothermal case in a bounded domain Ω ⊂ ℝ³:

where U : Ω ⊂ ℝ³ → ℝ³ is the fluid velocity and ρ:Ω ⊂ ℝ³ → ℝ is its density, meanwhile, it satisfies ρ ≥ 0. Moreover, the total mass is described by

The constant viscosity coefficients μ and λ are assumed to satisfy μ > 0 and μ+λ > 0. P(ρ) denotes the pressure, it satisfies P(ρ) = ρ for the isothermal case. f : Ω → ℝ³ is a given function which models an outer force density. For simplicity we assume that the domain Ω is an unit ball with a radius 1 and smooth boundary ∂Ω, namely,

We treat the case of Dirichlet boundary conditions

The spherically symmetric solution (ρ, U) of the problem (1.4) enjoys the form

On one hand, in the spherical coordinates, the original system (1.4) under the assumption (1.7) takes the form

then mutiplying the first equation of (1.8) by u, we derive

by which, we reduce the second equation of (1.8) into

On the other hand, we multiply the first equation of (1.8) by ρ u to get

which gives that

where C is an arbitrary constant (We will set it to be a big positive constant). Assume that u(r) > 0, then we have

We substitute (1.10) into the second equation of (1.9), then we obtain the singular quasi-linear ODEs (1.1). Meanwhile, the total mass (1.5) in spherically symmetric case is equivalent to the constraint condition (1.3). Let the pressure law be P(ρ) = ρ^γ in (1.4), when the adiabatic exponent γ > 53 , the first existence of weak solution for the system (1.4) was given by Lions [11]. Novotný-Strašcraba employed the concept of oscillation defect measure developed in [4] to prove the existence result for all γ > 32 . Frehse-Steinhauer-Weigant [7] got the existence of weak renormalized solutions to problem (1.4) for all γ > 43 . After that, Plotnikov-Weigant [16] extended the exstence of weak solution of problem (1.4) to the case of γ > 1. Feireisl-Novotný [5] studied the existence of weak solutions with arbitrarily large boundary data for stationary compressible Navier-Stokes system with general inhomogeneous boundary conditions. Meanwhile, they required the pressure to be given by the standard hard sphere EOS.

When the adiabatic exponent γ = 1, in two dimensional case, Lions [11] proved the existence of weak solution for this stationary system by means of a slightly modified equation of mass conservation aρ + div (ρ u) = 0. After that, Frehse-Steinhauer-Weigant [6] improved this result. In the three dimension case, Lions [11] pointed out that it is an open problem. To our knowledge, there is no result for the three dimensional viscous compressible isothermal stationary Navier-Stokes equations. In this paper, we are devoted to solving this problem by construction of Sobolev regularity solutions for problem (1.4) with the spherically symmetric case.

Notations.

Thoughout this paper, let Ω = (0, 1]. we denote the usual norm of 𝕃²(Ω) and Sobolev space H^s(Ω) by ∥⋅∥_𝕃² and ∥⋅∥_H^s, respectively. The symbol a ≲ b means that there exists a positive constant C such that a ≤ Cb. The letter C with subscripts to denote dependencies stands for a positive constant that might change its value at each occurrence.

2 Proof of Theorem 1.1

In order to solve the dissipative quasi-linear ODE (1.1) with the boundary condition (1.2), we should overcome two difficulties:

1. There is the loss of derivative phenomenon. It means that the classical fixed point theorem can not be used.

2. A positive solution u(r) should be constructed to satisfy the condition (1.3).

Hence, we have to construct a H^s(Ω)-solution for the dissipative quasi-linear ODE (1.1) with the boundary condition (1.2) by using a suitable Nash-Moser iteration scheme. This method has been used in [18, 19, 20, 21, 22, 23]. For general Nash-Moser implict function theorem, one can see the celebrated work of Nash [13], Moser [12] and Hörmander [9], and Rabinowitz [17].

We now introduce a family of smooth operators possessing the following properties.

In our iteration scheme, we set

Then, by (2.1), it holds

We consider the approximation problem of dissipative quasi-linear ODE (1.10) as follows

We first show how to construct positive smooth solution for the dissipative quasi-linear ODE (1.1) with the zero boundary condition, then by means of some transformation, the non-zero boundary condition case can be transformed into the zero boundary case (see page 15). One can see [1, 3, 10, 14] for more related results on elliptic-type equations.