Boundary layer theory for convection-diffusion equations in a circle (Q2925749)

This paper is dedicated to the memory of M. Vishik. It deals with the asymptotic behavior, as \(\epsilon\to 0\), of the solution to problem \((P_{\epsilon}):\) NEWLINE\[NEWLINEL_{\epsilon}u^{\epsilon}=-\epsilon \Delta u^{\epsilon}-u^{\epsilon}_y=f \text{ in }D,~u^{\epsilon}=0 \text{ on }\partial D,NEWLINE\]NEWLINE where \(D\) is the unit disk with center \((0,0)\), and \(f\) is a sufficiently smooth function. The limit problem \((P_{o})\) is defined by NEWLINE\[NEWLINE-u_y^{o}=f \text{ in } D,~ u^{o}=0 \text{ on } \Gamma_u=\{(x,y);x^2+y^2=1,~y>0\}.NEWLINE\]NEWLINE Two cases were studied by the authors in previous papers, namely: the compatible case i.e. \(f\) is sufficiently flat at the characteristic points \((\pm 1,0),\) in [J. Math. Pures Appl. (9) 96, No. 1, 88--107 (2011; Zbl 1228.35021)], and the non-compatible case i. e. \(f\) is polynomial, in [SIAM J. Math. Anal. 44, No. 6, 4274--4296 (2012; Zbl 1261.35013)]. These previous results are recalled in Sections 2 and 3 of the present paper. Next, the general case is studied. For that, the right member of \((P_{\epsilon})\) is written on the form: NEWLINE\[NEWLINEf=f-\hat{f}+\rho(x)\check{\rho}(x)\hat{f}+(1-\rho(x)\check{\rho}(x))\hat{f},NEWLINE\]NEWLINE where \(\hat{f}\) is the Taylor expansion of \(f\) to some order and \(\rho,\check{\rho}\) are smooth cut-off functions vanishing identically in a small neighbourhood of \((1,0)\) and \((-1,0)\), respectively (\(\check{\rho}(x)=\rho(-x)\)). Thanks to the above previous results and the linearity of problem \((P_{\epsilon})\), it is sufficient to concentrate on the case where \(f\) is replaced by \((1-\rho(x)\check{\rho}(x))\hat{f}\). This case produces the new parabolic boundary layers . It is pointed out that the corresponding boundary layer equations cannot be solved explicitly. Nevertheless, the authors obtain various estimates of the boundary layer functions, by means of suitable adapted kernels denoted by \(M_{l,d}\) (Lemma 8).