arXiv subjects

Charis Tsikkou

Publications and source records attributed to Charis Tsikkou.

15 recordsLinked to original sources

arXiv2026

The Riemann Problem for a 3x3 Generalized Chaplygin Gas System with Variable Pressure

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$, $0<\alpha\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $\delta$-shock, we assume $\alpha\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $\delta$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.

math.AP↗

arXiv2026

Dynamic Response of a Finite Circular Plate on an Elastic Half-Space Using the Truncated Lamb Kernel

We develop an exact operator formulation for the dynamic interaction between a finite circular elastic plate and an elastic half-space. Classical analyses, beginning with Lamb's representation of the half-space response, typically assume an infinite plate and rely on diagonalization of the soil operator via the continuous Hankel transform. For a plate of finite radius $R$, however, both traction and displacement are supported only on $0 \le r \le R$, leading to the spatially truncated Lamb operator \[ \mathscr{M}(\omega) = \chi_{[0,R]} \, T(\omega)\, \chi_{[0,R]}, \] where $T(\omega)$ is the Hankel multiplier involving the Rayleigh denominator $\Omega(\xi,\omega)$. Truncation destroys the diagonal structure of $T(\omega)$ and introduces real-axis singularities associated with the Rayleigh pole, in addition to square-root branch points at $\xi = k_T$ and $\xi = k_L$. We represent the action of $\mathscr{M}(\omega)$ on a finite-disk Bessel basis $\{ \phi_n(r) = A_{1,n} J_0(\lambda_n r) + A_{2,n} I_0(\lambda_n r)\},$ which satisfies the free-edge boundary conditions of the plate, and derive explicit expressions for the resulting matrix elements. These involve integrals of the Lamb kernel evaluated as Cauchy principal values, with residue contributions corresponding to radiation damping in the half-space. The resulting operator matrix is dense but spectrally convergent. Its inversion yields a complete frequency-domain solution for finite-radius plates. The analysis reproduces Chen et al.'s finite-radius experiments for small $R$, approaches the infinite-radius limit as $R \to \infty$, and quantifies finite-radius corrections. To our knowledge, this is the first exact operator-level treatment of finite-radius plate-half-space interaction that retains the full nonlocal Lamb kernel.

math.AP↗

arXiv2025

Non-isentropic cavity flow for the multi-d compressible Euler system

We rigorously construct non-isentropic and self-similar multi-d Euler flows in which a central cavity (vacuum region) collapses. While isentropic flows of this type have been analyzed earlier by Hunter \cite{hun_60} and others, the non-isentropic setting introduces additional complications, in particular with respect to the behavior along the fluid-vacuum interface. The flows we construct satisfy the physical boundary conditions: the interface is a material surface along which the pressure vanishes, and it propagates with a non-vanishing and finite acceleration until collapse. We introduce a number of algebraic conditions on the parameters in the problem (spatial dimension, adiabatic index, similarity parameters). With these conditions satisfied, a simple argument based on trapping regions for the associated similarity ODEs yields the existence of non-isentropic cavity flows. We finally verify that the conditions are all met for several physically relevant cases in both two and three dimensions.

math.AP↗

arXiv2025

An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term

We consider a system consisting of one conservation law and one balance law with a time-dependent source term, and provide a comprehensive analysis of Riemann solutions, including the non-classical overcompressive delta shocks. The minimal yet representative structure of the system captures essential features of transport under density constraints and, despite its simplicity, serves as a versatile prototype for crowd-limited transport processes across diverse contexts, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. In addition to non-self-similar solutions mentioned above, the associated Riemann problem admits solution structures that traverse vacuum states ($\rho = 0$) and the critical density threshold ($\rho = \bar{\rho}$), where mobility vanishes and characteristic speed degenerates. Moreover, the explicit time dependence in the source term leads to the breakdown of self-similarity, resulting in distinct Riemann solutions over successive time intervals and highlighting the dynamic nature of the solution landscape. The theoretical findings are numerically confirmed using the Local Lax-Friedrichs scheme.

math.AP↗

arXiv2025

An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Conservation Laws Using Shadow Waves and Dafermos Regularization

We consider a system of two conservation laws and provide a detailed description of both classical and non-classical self-similar Riemann solutions. In particular, we demonstrate the existence of overcompressive delta shocks as singular limits of the Dafermos regularization of the system. The system is chosen for its minimal yet representative structure, which captures the essential features of transport dynamics under density constraints. Our analysis is carried out using blow-up techniques within the framework of Geometric Singular Perturbation Theory (GSPT), allowing us to resolve the internal structure of these singular solutions. Despite its simplicity, the system serves as a versatile prototype for crowding-limited transport across a range of applications, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. Our findings are supported by numerical simulations using the Local Lax-Friedrichs scheme.

math.AP↗

arXiv2024

Solving the Wave Equation on Discrete Time Scales

This paper presents a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is set to the integers and a broader class of discrete time scales, while the time dimension is set to the positive real numbers.

math.AP↗

arXiv2024

An Analysis of a 2x2 Keyfitz-Kranzer Type Balance System with Varying Generalized Chaplygin Gas

We consider a system of two balance laws of Keyfitz-Kranzer type with varying generalized Chaplygin gas, which exhibits negative pressure and is a product of a function of time and the inverse of a power of the density. The Chaplygin gas is a fluid designed to accommodate measurements for the early universe and late-time universal expansion while obeying the pressure-density-time relation. We produce an explanation and description of the non-self-similar Riemann solutions, including the non-classical singular solutions. We also find that due to a direct dependence on time, a change in the regions allowing for combinations of classical and non-classical singular solutions occurs, therefore a Riemann solution can have different solutions over several time intervals. Our findings are confirmed numerically using the local Lax-Friederichs scheme.

math.AP↗

arXiv2022

Radially Symmetric Non-isentropic Euler flows: continuous blowup with positive pressure

Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions it is therefore unclear if it is the zero-pressure region that is responsible for blowup. The same applies to self-similar Euler flows describing radial cavity flow, first analyzed by Hunter (1960). Recent works have shown that the simplified isothermal and isentropic models admit continuous blowup solutions in the presence of a strictly positive pressure field. In this work we extend this conclusion to the case of the full Euler system. The solutions under consideration are radial self-similar flows in which a continuous wave focuses and blows up. We propagate the solutions beyond blowup and observe numerically that there are cases where an expanding spherical shock wave is generated at collapse. The resulting solution has the unusual property that the flow is isentropic in each of the two regions separated by the shock. We finally verify that these are admissible global weak solutions to the full, multi-d compressible Euler system.

physics.flu-dyn↗

arXiv2019

Multi-d Isothermal Euler Flow: Existence of unbounded radial similarity solutions

We show that the multi-dimensional compressible Euler system for isothermal flow of an ideal, polytropic gas admits global-in-time, radially symmetric solutions with unbounded amplitudes due to wave focusing. The examples are similarity solutions and involve a converging wave focusing at the origin. At time of collapse, the density, but not the velocity, becomes unbounded, resulting in an expanding shock wave. The solutions are constructed as functions of radial distance to the origin $r$ and time $t$. We verify that they provide genuine, weak solutions to the original, multi-d, isothermal Euler system. While motivated by the well-known Guderley solutions to the full Euler system for an ideal gas, the solutions we consider are of a different type. In Guderley solutions an incoming shock propagates toward the origin by penetrating a stationary and "cold" gas at zero pressure (there is no counter pressure due to vanishing temperature near the origin), accompanied by blowup of velocity and pressure, but not of density, at collapse. It is currently not known whether the full system admits unbounded solutions in the absence of zero-pressure regions. The present work shows that the simplified isothermal model does admit such behavior.

math.AP↗

arXiv2018

On similarity flows for the compressible Euler system

Radial similarity flow offers a rare instance where concrete inviscid, multi-dimensional, compressible flows can be studied in detail. In particular, there are flows of this type that exhibit imploding shocks and cavities. In such flows the primary flow variables (density, velocity, pressure, temperature) become unbounded at time of collapse. In both cases the solution can be propagated beyond collapse by having an expanding shock wave reflect off the center of motion. These types of flows are of relevance in bomb-making and inertial confinement fusion, and also as benchmarks for computational codes; they have been investigated extensively in the applied literature. However, despite their obvious theoretical interest as examples of unbounded solutions to the multi-dimensional Euler system, the existing literature does not address to what extent such solutions are bona fide weak solutions. In this work we review the construction of globally defined radial similarity shock and cavity flows, and give a detailed description of their behavior following collapse. We then prove that similarity shock solutions provide genuine weak solutions, of unbounded amplitude, to the multi-dimensional Euler system. However, both types of similarity flows involve regions of vanishing pressure prior to collapse (due to vanishing temperature and vacuum, respectively) - raising the possibility that Euler flows may remain bounded in the absence of such regions.

math.AP↗

arXiv2017

Two Dimensional Riemann Problems for the Nonlinear Wave System: Rarefaction Wave Interactions

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

math.AP↗

arXiv2016

On convergence of exterior solutions to radial Cauchy solutions for $\square_{1+3}U=0$

Consider the Cauchy problem for the 3-d linear wave equation $\square_{1+3}U=0$ with radial initial data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. A standard result gives that $U$ belongs to $C([0,T];H^s(\mathbb{R}^3))$ whenever $(Φ,Ψ)\in H^s\times H^{s-1}(\mathbb{R}^3)$. In this note we are interested in the question of how $U$ can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls $B_\varepsilon$ about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of $H^s$ well-posedness for the wave equation. We show how explicit solution formulae yield convergence and optimal regularity for the Cauchy solution via exterior solutions, when the latter are extended continuously as constants on $B_\varepsilon$ at each time. We establish that for $s=2$ the solution $U$ can be realized as an $H^2$-limit (uniformly in time) of exterior solutions on $\mathbb{R}^3\setminus B_\varepsilon$ satisfying vanishing Neumann conditions along $|x|=\varepsilon$, as $\varepsilon\downarrow 0$. Similarly for $s=1$: $U$ is then an $H^1$-limit of exterior solutions satisfying vanishing Dirichlet conditions along $|x|=\varepsilon$.

math.AP↗

arXiv2015

Radial solutions to the Cauchy problem for $\square_{1+3}U=0$ as limits of exterior solutions

We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the 3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric Cauchy problem. In treating the 3-d wave equation we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3-d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in $C([0,T);H^s(\mathbb{R}^3))$ whenever $(Φ,Ψ)\in H^s\times H^{s-1}(\mathbb{R}^3)$. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in $H^2\times H^1(\mathbb{R}^3)$ we obtain $H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via exterior Dirichlet solutions.

math.AP↗

arXiv2015

Singular Shocks in a Chromatography Model

We consider a system of two equations that can be used to describe nonlinear chromatography and produce a coherent explanation and description of the unbounded solutions (singular shocks) that appear in M. Mazzotti's model. We use the methods of Geometric Singular Perturbation Theory, to show existence of a viscous solution to Dafermos-DiPerna regularization.

math.AP↗

arXiv2014

Analysis of 2+1 diffusive-dispersive PDE arising in river braiding

We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - γu_{xxx} -αu_{yyyy} - βu_{yy} + \left (u^2 \right)_{xyy} = 0$$ for $(x,y) \in [0, 2π] \times [0, π]$, $t> 0$, with boundary condition $u_{y}=0=u_{yyy}$ at $y=0$ and $y=π$ and $2π$ periodicity in $x$, using a contraction mapping argument in a Bourgain-type space $T_{s,b}$. We also show that the energy $\| u \|^2_{L^2} $ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.

math.AP↗