Frozen Lopatinski symbols and exact spherical capillary mode corrections

Definitions and mathematical statement

Question and recovered context. Determine the relation between the stability Lopatinski determinant and spherical-harmonic analyses of spherical waves, including the work of C. C. Wu and P. H. Roberts on bubble shape instability. The supplied verification places the question under S. Benzoni-Gavage, H. K. Jenssen and M. Williams, "Existence and Stability of Spherical Fronts", concerning curved shock, reactive or viscous fronts. It supplies the preceding radial barotropic equations in dimensions $d=2,3$: \[ \rho_t+(\rho u)_r+\frac{(d-1)\rho u}{r}=s(r,t), \] \[ (\rho u)_t+(\rho u^2+P(\rho))_r+\frac{(d-1)\rho u^2}{r} =\nu\left(u_r+\frac{(d-1)u}{r}\right)_r+F(r,t). \] Here $u$ is radial velocity, $P$ the pressure law, and $s,F$ source terms. Their values, base front, boundary/radiation conditions and admissible perturbations are not specified. The verification says an earlier question concerns stationary fronts and curvature/sources; its description of a subsequent question is truncated and cannot be reconstructed.

Reported frozen shock-symbol relation. For an isotropic inviscid planar Lax shock, the aid distinguishes its frozen principal Lopatinski symbol $\Delta_{\mathrm{Lop}}(\lambda,k)$ from a full spherical-front determinant. On a three-dimensional sphere of radius $R$, spherical harmonics satisfy \[ -\Delta_{S_R^2}Y_{\ell m}=\frac{\ell(\ell+1)}{R^2}Y_{\ell m},\qquad k_\ell=\frac{\sqrt{\ell(\ell+1)}}{R}. \] The reported frozen principal multiplier evaluates the planar symbol at $k=k_\ell$. Radial evolution, lower-order curvature, sources and base-state terms remain in the full front problem, so this substitution is not an exact identity of complete stability determinants. A viscous Evans function or a reactive-front determinant would require its own setup.

Separate static capillary calculation reported by the aid. For a static inviscid incompressible two-fluid capillary sphere in three dimensions, let $R>0$, surface tension $\sigma>0$, interior density $\rho_-\geq0$ and exterior density $\rho_+>0$. In the standard interior-regular/exterior-decaying potential-flow interpretation, with no imposed base flow or gravitational term, the supplied modal dispersion determinant for $\ell\geq1$ is \[ D_{\mathrm{sph},\ell}(\lambda)= \lambda^2R\left(\frac{\rho_-}{\ell}+\frac{\rho_+}{\ell+1}\right) +\frac{\sigma(\ell-1)(\ell+2)}{R^2}. \] The stated spatial/radiation interpretation makes the displayed formula meaningful; it is not supplied as boundary data for the original shock problem. The degree-zero volume mode is not covered, and $\ell=1$ is the neutral translation mode. With planar normalization \[ D_{\mathrm{pl}}(\lambda,k)=(\rho_-+\rho_+)\lambda^2+\sigma k^3, \] the aid's exact correction identity is \[ \begin{split} k_\ell D_{\mathrm{sph},\ell}(\lambda)-D_{\mathrm{pl}}(\lambda,k_\ell) ={}&\lambda^2\left[\rho_-\left(\sqrt{\frac{\ell+1}{\ell}}-1\right) +\rho_+\left(\sqrt{\frac{\ell}{\ell+1}}-1\right)\right]\\ &-\frac{2\sigma k_\ell}{R^2}. \end{split} \] The first correction comes from the spherical Dirichlet-to-Neumann factors and the second from mean curvature. The difference tends to zero in the joint limit $R,\ell\to\infty$, $k_\ell\to k>0$, for fixed $\lambda$ and fixed densities and surface tension. It is not an exact finite-radius planar substitution.

For the gas-bubble limit $\rho_-=0$, writing $\lambda=\mathrm i\omega_\ell$ for oscillatory modes, the aid further reports \[ \frac{\omega_\ell^2}{\sigma k_\ell^3/\rho_+} =1+\frac{1}{2\ell}+O(\ell^{-2})\qquad(\ell\to\infty). \] This capillary determinant is neither a shock Lopatinski determinant nor an Evans function. Its identity and asymptotic calibration do not resolve the full curved-front comparison for the unspecified sourced or viscous equations.

Short English statement

The aid distinguishes a frozen planar shock symbol evaluated at spherical-harmonic frequencies from a complete spherical-front stability problem. It also gives an exact but separate static capillary dispersion correction and its planar limit. The capillary formula is not identified with a shock or viscous stability determinant.

Sources

• Local UnsolvedMath registry, permanent ID $20002438$, catalogue entry, aim-pdes-notes.json, notes[49], workshop Stability criteria for multi-dimensional waves and patterns, open problem 2: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf.
• Supplied preceding-source attribution: S. Benzoni-Gavage, H. K. Jenssen and M. Williams, "Existence and Stability of Spherical Fronts".
• C. C. Wu and P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998), as explicitly cited by the workshop.
• Supplied research summary on frozen principal Lopatinski multipliers and the separate capillary modal determinant. Attempt 1: partial_result, partially_solved, rigor "proved", low novelty confidence. This machine-generated aid is not peer reviewed and requires independent verification before citation. The supplied verification reports a PDF comparison, but this draft has not visited it.