# Maria Radosz's research while affiliated with University of Texas at San Antonio and other places

## Publications (22)

Article
This paper considers the relativistic motion of charged particles coupled with electromagnetic fields in the higher-order theory proposed by Bopp, Landé–Thomas, and Podolsky. We rigorously derive a world line integral expression for the self-force of the charged particle from a distributional equation for the conservation of four-momentum only. Thi...
Article
Full-text available
New constraints are found that must necessarily hold for Israel-Stewart-like theories of fluid dynamics to be causal far away from equilibrium. Conditions that are sufficient to ensure causality, local existence, and uniqueness of solutions in these theories are also presented. Our results hold in the full nonlinear regime, taking into account bulk...
Preprint
Full-text available
We consider equations of M\"uller-Israel-Stewart type describing a relativistic viscous fluid with bulk viscosity in four-dimensional Minkowski space. We show that there exists a class of smooth initial data that are localized perturbations of constant states for which the corresponding unique solutions to the Cauchy problem breakdown in finite tim...
Chapter
The α-patch model is used to study aspects of fluid equations. We show that solutions of this model form singularities in finite time and give a characterization of the solution profile at the singular time.
Preprint
Full-text available
New constraints are found that must necessarily hold for Israel-Stewart-like theories of fluid dynamics to be causal far away from equilibrium. Conditions that are sufficient to ensure causality, local existence, and uniqueness of solutions in these theories are also presented. Our results hold in the full nonlinear regime, taking into account bulk...
Preprint
Nonlinear dynamic systems can be classified into various classes depending on the modeled nonlinearity. These classes include Lipschitz, bounded Jacobian, one-sided Lipschitz (OSL), and quadratically inner-bounded (QIB). Such classes essentially yield bounding constants characterizing the nonlinearity. This is then used to design observers and cont...
Preprint
The $\alpha$-patch model is used to study aspects of fluid equations. We show that solutions of this model form singularities in finite time and give a characterization of the solution profile at the singular time.
Preprint
This paper considers the relativistic motion of charged particles coupled with electromagnetic fields in the higher-order derivative theory proposed by Bopp, Land\'e--Thomas, and Podolsky. We rigorously derive a world-line integral expression for the self-force of the charged particle from a distributional equation for the conservation of four-mome...
Preprint
This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coeff...
Article
Full-text available
Córdoba et al. (Ann Math 162(3):1377–1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for whi...
Article
We investigate a system of nonlocal transport equations in one spatial dimension. The system can be regarded as a model for the 3D Euler equations in the hyperbolic flow scenario. We construct blowup solutions with control up to the blowup time.
Article
Full-text available
We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a simplified vorticity stretching term and Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable,...
Article
Full-text available
This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coeff...
Article
In this paper we study the singularity formation for two nonlocal 1D active scalar equations, focusing on the hyperbolic flow scenario. Those 1D equations can be regarded as simplified models of some 2D fluid equations.
Article
Full-text available
We consider smooth, double-odd solutions of the two-dimensional Euler equation in $[-1, 1)^2$ with periodic boundary conditions. It is tempting to think that the symmetry in the flow induces possible double-exponential growth in time of the vorticity gradient at the origin, in particular when conditions are such that the flow is "hyperbolic". This...
Article
Full-text available
We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting mod...
Data
Full-text available
We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting mod...
Article
We show a new limiting absorption and a new limit amplitude principle for periodic operators acting on functions defined on the whole $${\mathbb{R}^{d}}$$ . For a differential operator $${\mathcal{L}}$$ with periodic coefficients, we show the existence of the distributional limiting absorption solution of the Helmholtz equation (\({\mathcal{L}-\ome...

## Citations

... The run-away solutions can be avoided by using the critical submanifold [9], but this leads to pre-acceleration, where a particle moves before a force is applied. Alternative approaches include using a delay equation [3], the Eliezer-Ford-O'Connell equation [10], considering the Landau-Lifshitz equation as fundamental (and not an approximation) [11], variable mass [12,13], and replacing Maxwell's equation with Born-Infield [14] or Bopp-Podolski [15][16][17][18][19][20]. ...
... Initial studies have done this using a phenomenological Israel-Stewart approach in [34], and based on DNMR in [35] (although not with all possible DNMR terms). However, there have not yet been studies that can systematically connect the Israel-Stewart approach and DNMR, nor have there been causality and stability analyses (a la [36][37][38]) applied to BSQ diffusion yet. Here we use a new approach that can reconcile nearly all terms in the equation of motion between phenomenological Israel-Stewart and DNMR. ...
... The OSL constant can be computed by utilizing a derivative-free intervalbased global optimization algorithm according to [18]. This algorithm splits the set into smaller subsets and removes the ones that positively do not contain any maximizers, which reduces complexity. ...
... There has been a lot of effort in studying potential singularity of the 3D Euler equations using various simplified models. In [9,27,29,37], the authors proposed several simplified models to study the Hou-Luo blowup scenario [40,41] and established finite time blowup of these models. In these works, the velocity is determined by a simplified Biot-Savart law in a form similar to the key lemma in the seminal work of Kiselev-Sverak [36]. ...
... There has been a lot of effort in studying potential singularity of the 3D Euler equations using various simplified models. In [9,27,29,37], the authors proposed several simplified models to study the Hou-Luo blowup scenario [40,41] and established finite time blowup of these models. In these works, the velocity is determined by a simplified Biot-Savart law in a form similar to the key lemma in the seminal work of Kiselev-Sverak [36]. ...
... The C 1 2 conjecture is then simply to prove that solutions to the Cordoba-Cordoba-Fontelos model remain uniformly bounded in C 1 2 up to the time of blow-up (see [7,10,16,18,24]). We close by mentioning a recent work by Hoang and Radosz [13] which rigorously established cusp formation in a non-local model, which is a slight variant of the CCF model. ...
... Moreover, it is also of interest to understand the nature and location of the defect spectrum. For previous works regarding line defects in bandgap crystals we refer to [11,23,24,13,14,15,16,18]. ...
... One interesting aspect is that the equation has similarities to one-dimensional transport equations that are studied in fluid dynamics, see e.g. [9,10,11,12,14,18,20,38,39,51]. Granero-Belinchon [29] studied an analogue of the equation on the one-dimensional torus. ...
... In the direction of improving the upper bounds, in [15], the authors showed that for the incompressible Euler equations in the torus, if the vorticity is initially C 2 and enjoys a double reflection symmetry, under some restrictive conditions, the solution admits at most exponential growth for the gradient of vorticity in a small region near the origin. ...
... Estimate (19) follows now from (20). ...