Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01690

Nonlinear stability of the Guderley imploding shock wave in radial symmetry

Giorgio Cialdea
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove nonlinear stability of the Guderley imploding shock under radial perturbations for the three-dimensional compressible Euler equations in the adiabatic range $1<γ\leq\frac{5}{3}$. The perturbed solutions develop an asymptotically self-similar implosion in finite time. Our result allows perturbations with small, nonzero constant pressure in the quiescent region ahead of the shock. The proof combines weighted $L^\infty$ estimates along characteristics with a computer-assisted proof of stability inequalities for the Guderley profile.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giorgio Cialdea. 2026-10-01. Nonlinear stability of the Guderley imploding shock wave in radial symmetry. https://arxiv.org/abs/2610.01690

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

We extend the De Giorgi--Nash--Moser theory to nonlocal hypoelliptic equations arising in kinetic theory as linearized models for the non-cutoff Boltzmann equation. Assuming that the nonlocal tail in velocity of weak solutions belongs locally to $L^p_{t,x}$ for some $p>N_{d}/(2s)$, where $N_{d}$ is the drift homogeneous dimension and $2s\in(0,2)$ is the order of diffusion, we prove a local $L^2$-$L^\infty$ estimate, from which we also deduce a nonlocal strong Harnack inequality. The tail summability threshold for the local estimate is optimal: for every $1\leq p \leq N_{d}/(2s)$, we construct a family of nonnegative, time-dependent solutions that rules out this estimate even for the pure fractional Kolmogorov equation. The local boundedness estimate also allows for possibly unbounded source terms.

math.AP↗
arXiv2026

Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_p)^s u = λ\frac{f(u)}{u^β}\ \text{in} \ Ω\newline u >0\ \text{in} \ Ω,\ u =0\ \text{in} \ \mathbb{R}^N \setminus Ω\end{aligned} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $λ> 0 $ is a parameter, $0\leq β<1$ and $f$ is a non-decreasing $C^1$-function with $f(0)>0$. Using the method of sub- and supersolutions, we present a novel multiplicity result and, in specific cases, we also prove a three-solution theorem using Amann's fixed point theorem. Our construction of sub-supersolutions avoids the conventional reliance on ODE techniques and Green's function estimates, thereby making it more adaptable to the nonlinear and nonlocal framework. Additionally, we establish a Hopf-type Strong Comparison Principle for the linear operator with singular nonlinearity, marking the first result of its kind for mixed local-nonlocal operators. This result is crucial in deriving a third solution and holds broader mathematical significance.

math.AP↗
arXiv2026

The Obstacle Problem Arising from the American Chooser Option

We study the finite-horizon obstacle problem associated with the American chooser option. Its obstacle is the maximum of two independently determined American call and put values and has a spatial kink where these values cross. Using truncation, penalization, and comparison, we construct a local strong Sobolev solution and prove uniqueness within an exponential-growth class. We show that contact occurs only on the smooth exercise-payoff branches and that both contact regions have nonempty sections at every time. We then establish strict monotonicity, terminal limits, and local Lipschitz continuity of the two free boundaries. Localized one-phase equations, quantitative nondegeneracy, and parabolic boundary Harnack estimates yield $C^\infty$ regularity on the open time interval.

math.AP↗