Sobolev regularity solutions for a class of singular quasilinear ODEs

: This paper considers an initial-boundary value problem for a class of singular quasilinear second-order ordinary differential equations with the constraint condition stemming from fluid mechanics. We prove that the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs by means of a suitable Nash-Moser iteration scheme. Meanwhile, asymptotic expansion of those positive solutions is shown.


Introduction and main results
In this paper, we consider the following singular second-order ordinary di erential equations ur + r u + C − ( µ + λ)r urr + r( µ + λ) − C ur − ( µ + λ)u u + uf (r) = , r ∈ ( , ], (1.1) a bounded domain Ω ⊂ R : − µ U − (λ + µ)∇divU + div(ρU ⊗ U) + ∇P(ρ) = ρf , div(ρU) = , (1.4) where U : Ω ⊂ R → R is the uid velocity and ρ : Ω ⊂ R → R is its density, meanwhile, it satis es ρ ≥ . Moreover, the total mass is described by The constant viscosity coe cients µ and λ are assumed to satisfy µ > and µ + λ > . P(ρ) denotes the pressure, it satis es P(ρ) = ρ for the isothermal case. f : Ω → R 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 and smooth boundary ∂Ω, namely, We treat the case of Dirichlet boundary conditions On the other hand, we multiply the rst equation of (1.8) by ρu to get d dr (ρu) + (ρu) r = , which gives that where C is an arbitrary constant (We will set it to be a big positive constant). Assume that u(r) > , then we have ρ = C r u . (1.10) 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 > , the rst 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 > . Frehse-Steinhauer-Weigant [7] got the existence of weak renormalized solutions to problem (1.4) for all γ > . After that, Plotnikov-Weigant [16] extended the exstence of weak solution of problem (1.4) to the case of γ > . 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 = , in two dimensional case, Lions [11] proved the existence of weak solution for this stationary system by means of a slightly modi ed equation of mass conservation aρ+div(ρu) = .
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 = ( , ]. we denote the usual norm of L (Ω) and Sobolev space H s (Ω) by · L 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.

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 di culties: 1. There is the loss of derivative phenomenon. It means that the classical xed point theorem can not be used. 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. Lemma 2.1. [2,8] Let Ω ⊂ R n with dimension n ≥ . There is a family {Π θ } θ≥ of smoothing operators in the space H s (Ω) acting on the class of functions such that We consider the approximation problem of dissipative quasi-linear ODE (1.10) as follows We rst 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.
. The rst approximation step m = Let constants s ≥ and < ε < . For any r ∈ Ω, we choose the initial approximation function u ( ) (r) ∈ H s+ (Ω). Meanwhile, for a xed small constant c(< ε), it satis es whereC is a postive constant, and E ( ) denotes the error term taking the form We now nd the rst approximation solution denoted by u ( ) of (2.3). The error step between the initial approximation function and rst approximation solution is denoted by then linearizing nonlinear system (2.3) around u ( ) to get the linearized operators as follows (2.5) We consider the linear system from which, the solution of it gives the rst approximation solution of dissipative quasi-linear ODE (1.1). Thus some priori estimates are needed. We rst give L -estimate of solution for (2.6).

Lemma 2.2. Let the initial approximation function u ( ) satisfy (2.4). Assume that f ∈ H (Ω) and f H ≤ . The solution h ( ) of the linear system (2.6) satis es
Proof. Multiplying both sides of the rst equation (2.6) by h ( ) and h ( ) r , respectively, it holds d dr and where we use We sum up (2.7) and (2.8) to get (2.10) (2.12) Note that the boundary condition given in (2.6). We integrate equality (2.9) over Ω, it holds On one hand, we notice that the rst approximation funciton u ( ) satisfy (2.4). So by (2.1), Young's inequality and f H ≤ , there exists a su cient big positive constant C such that where C ε,C is a positive constant depending on ε and C, it will be small as ε small. On the other hand, by Young's inequality, it holds Thus, by (2.14)-(2.15), it follows from (2.13) that Note that r ∈ ( , ] and constant C ε,C being small as ε small. Thus there exists a positive constant C such that − C ε,C ≥ C > , Hence, it follows from (2.16) that Furthermore, we derive higher order derivatives estimates. For a xed integer s ≥ , we apply D s := d s dr s to both sides of (2.6), it holds with the boundary condition where the integer ≤ k ≤ s, and Here we notice this term s i +i =s, ≤i ≤s− (D i ( r ))D i h ( ) can cause some troubles when we carry out energy estimates due to s i +i =s, ≤i ≤s− Next we derive higher derivative estimate of solution for (2.6).

Lemma 2.3.
Let the initial approximation function u ( ) satisfy (2.4). Assume that f ∈ H s (Ω) and f H s ≤ for any xed s ≥ . The solution h ( ) of the linear system (2.6) satis es Proof. This proof is based on the induction. Let s = , by (2.17)-(2.18), it holds Note that the boundary condition (2.20). We integrate equality (2.21) over Ω, it holds We now estimate each term in the right hand side of (2.22). Let constant < ε . Since we have chosen the rst approximation funciton u ( ) satisfying (2.4), by Young's inequality and (2.1), f H ≤ and r ∈ ( , ], for a su cient big C, it holds A (r) L ∞ ≤ C ε,C , where C ε,C is a positive constant depending on ε and C, which can be small as ε small. Thus by (2.23), we derive (2.24) On the other hand, by Young's inequality, we integrate by part to get Hence, by (2.24)-(2.25), we can reduce (2.22) into (2.26) Note that r ∈ ( , ] and constant C ε,C being small as ε small. Thus there exists a positive constant C such that − C ε,C ≥ C > , which combining with (2.26) gives that Assume that the (s − )th derivative case holds, i.e., We now prove the sth derivative case holds. Obviously, equation (2.17) can be written as (2.29)
Thus we derive On one hand, we use (2.35) and Young's inequality to derive s i +i =s, ≤i ≤s−  (2.37) Note that r ∈ ( , ] and constant C ε,C being small as ε small. Thus there exists a positive constant C such that − C ε,C ≥ C > , Therefore, by (2.37), we obtain (D s+ h ( ) ) + (D s h ( ) ) dr s k= (D k E ( ) ) dr.
where G (r) = , E ( ) T and the matrix A(r) is where the matrix A − (r) is the inverse of matrix A(r) due to r (u ( ) ) > by (2.4), and the coe cients Following [24], by the standard xed point iteration and a priori estimate in Lemma 2.3, we obtain the approximation problem has a Cauchy sequence {v ( ) j (r)} j∈Z + in H s (Ω) for s ≥ , whose limit is v ( ) (r), and it solves the linear problem (2.6) in ( , ]. Furhermore, summing up both estimates given in Lemma 2.2-2.3, one can derive the estimate (2.38).

. The mth approximation step
Let R ∈ ( , ) be a xed constant. We de ne with the integers ≤ k ≤ m − and s ≥ . Assume that the m-th approximation solutions of (2.