Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries

In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ ( ρ ) = ρ θ , λ ( ρ ) = ( θ − 1) ρ θ in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> RN . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ. , vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>θ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> θ =2/3 , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>γ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>5</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> γ =5/3 , respectively.