Non-radiative Case Research Articles

The influence of baryons on low-mass haloes

ABSTRACT The Voids-within-Voids-within-Voids project used dark-matter-only (DMO) simulations to study the abundance and structure of dark matter (DM) haloes over the full mass range populated in the standard Lambda cold dark matter cosmology. Here, we explore how baryonic effects modify these results for $z=0$ halo masses in the range $10^4$–$10^7~\mathrm{M_\odot }$, below the threshold for galaxy formation. Our main study focuses on three simulations from identical initial conditions at $z=127$, one following DMO, one including non-radiative gas, and one additionally including the baryonic physics relevant in this halo mass range (cooling and photoheating). In the non-radiative simulation, above $10^{5.5}~\mathrm{M_\odot }$, halo abundance and internal structure are very similar to the DMO simulation, and the baryon to DM ratio is everywhere close to the cosmic value. At lower mass, this ratio drops and haloes are less concentrated and less massive in the non-radiative case. Test simulations at higher resolution show this to be mainly a resolution effect; the expected drop in baryon content due to residual pressure effects only becomes substantial for $z=0$ haloes below ${\sim}10^{2.7}~\mathrm{M_\odot }$. However, gas is heated by reionization at $z=6$ in our ‘full physics’ run, and this results in almost complete expulsion of gas from all haloes in our simulated mass range. This suppresses the halo mass function by ${\sim}30{{\ \rm per\ cent}}$, lowers halo concentration, and consequently weakens the DM annihilation signal by ${\sim}40{-}60{{\ \rm per\ cent}}$.

Local and global properties of p-Laplace Hénon equation

In this study, we begin by providing apriori estimates for positive radial solutions of the p-Laplace Hénon equation. Subsequently, we investigate the local and global properties of these solutions. Finally, we extend some of the radial results to the non-radial case.

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$\begin{aligned} i \partial _t u +\Delta u +|x|^{-b} |u|^{2\sigma }u = 0, \,\,\, x \in {\mathbb {R}}^N \end{aligned}$$with \(N\ge 3\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\) satisfies \(0<s_c<1\). In a previous work, for radial initial data in \(\dot{H}^{s_c}\cap \dot{H}^1\), we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in \(H^1\).

Emergence of large densities in a chemotaxis system with signaling loops, nonlinear signal productions and competitions sources under nonradial symmetry case

Emergence of large densities in a chemotaxis system with signaling loops, nonlinear signal productions and competitions sources under nonradial symmetry case

Scattering of the focusing energy-critical NLS with inverse square potential

We consider the Cauchy problem for the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential in dimension $ d = 4, 5, 6 $. We show that if the supremum of the kinetic energy of a solution over its maximal lifespan is less than the kinetic energy of the ground state, then the solution must exist globally in time and scatter in both time directions. We develop the 'long-term kinetic energy decoupling' associated with the appearance of inverse square potential. No radial assumption is made on the initial data. This extends the result in [22] by the first author to the non-radial case.

On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement II. The non-radial case

Abstract In this article, we study uniqueness and nonuniqueness for internal potential reconstruction from one boundary measurement in quantum fields that is related to the steady state Schrödinger equation. It is an extension of our recent work [On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement, J. Comput. Appl. Math. 381 (2021), Paper No. 113029]. Based on the theory of the ND map and modified Bessel function, the uniqueness theorem of the inverse problem in two-dimensional and three-dimensional core-shell structure is established, respectively. When different internal potential and shape are considered, the nonuniqueness results are also proved. We then present some numerical examples to illustrate our theoretical results.

Perturbed Fourier uniqueness and interpolation results in higher dimensions

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska [1].

Influence of thermal radiation on layered dust explosions

Multidimensional unsteady numerical simulations were carried out to explore the influence of thermal radiation on the propagation and structure of layered coal dust explosions. The simulation solved the reactive compressible Navier-Stokes equations coupled to an Eulerian kinetic-theory-based granular multiphase model. The radiation heat transfer is modeled by solving the radiation transfer equation using the third-order filtered spherical harmonics approximation. The radiation was assumed to be gray and all boundaries of the domain are black at 300 K. The reaction mechanism is based on global irreversible reactions for each physical process including devolatilization, char burning, moisture vaporization, and methane combustion. The governing equations were solved using a high-order Godunov method. Several simulation configurations were considered: layer volume fractions of 47% and 1%, channel lengths of 10 m and 40 m, and radiative and non-radiative cases. The results show that gray radiation has a significant influence on the propagation and structure of a layered dust explosion. However, radiation can have opposite effects on different scenarios. For example, radiation promotes the propagation of the dust flame when the layer volume fraction was 1% and in the short-channel cases where reflected shock-flame interactions are important. However, radiation enhances quenching for the 47% volume fraction dust layer in the longer channel.

Understanding effects of radiation from radiative shock on Richtmyer-Meshkov instability

&lt;sec&gt;Radiative shocks are ubiquitous in stellar environments and are characterized by high temperature plasma emitting a considerable fraction of their energy as radiation. Radiative shocks occur commonly in nature, especially in astronomical systems and inertial confinement fusion. The study of the effects of radiation on Richtmyer-Meshkov (RM) instability will improve our ability to understand and predict the evolution of RM instability under high energy density conditions.&lt;/sec&gt;&lt;sec&gt;A few experiments have been performed to compare the radiative case with the non-radiative case in Rayleigh-Taylor (RT) instability, thereby studying how the radiative effects change the evolution of RT instability, but the interplay between RM instability and radiative shock has been studied rarely. &lt;/sec&gt;&lt;sec&gt;This paper reports mainly the role of radiation in the changing of the RM instability. Two experiments are performed at Shenguang III prototype laser facility, the RM instability growth data are obtained by varying the laser intensity. The laser intensity for high-drive experiment is approximately 60% greater than that for low-drive experiment. The target consists of a multiple layer in the axial direction, in which the first layer is a 15μm-thick CH sample serving as an ablator, followed by a 10 μm-thick aluminum used as a shield layer to prevent the preheat effect. The next layer is a 350-μm-thick SiO&lt;sub&gt;2&lt;/sub&gt; foam, which is used as a material to produce a radiative shock. The last layer is the CH perturbed sample. There is a sinusoidal perturbation on the surface of CH sample which is adjacent to the SiO&lt;sub&gt;2&lt;/sub&gt; foam. The target is irradiated by four overlapping laser beams, and the laser beams produce a large pressure that drives a shock wave, whose velocity can be changed by varying the laser intensity, into the target package.&lt;/sec&gt;&lt;sec&gt;In the experiments, shock-generated radiative fluxes first ablate the unstable interface which the shock has not passed through, then the shock transmits the unstable interface to produce the RM instability. The images of unstable interface are captured using side-on x-ray radiography, and the experimental results show that the RM growth is suppressed in the experiment for the higher laser intensity. Radiation hydrodynamic code Multi1D is used to evaluate the electron temperature, shock velocity, and electron density. The simulations show that the foam temperature in the high-drive case can reach 80 eV in the front of shock, this energy flows away from the shock front, generating a radiative precursor ahead of the shock. The radiative precursor velocity of 270 km/s is much larger than the shock velocity of 170 km/s, the radiative precursor arrives at the unstable interface before the shock and ablates the unstable interface, so the radiative flux changes the initial conditions of unstable interface. When the shock propagates through the unstable interface, the ablation increases the density gradient length scale and reduces the Atwood number of the unstable interface, so the RM growth is suppressed in the high-drive case because of the ablation of the radiative precursor.&lt;/sec&gt;

Surface-Localized Transmission Eigenstates, Super-resolution Imaging, and Pseudo Surface Plasmon Modes

We present the discovery of a novel and intriguing global geometric structure of the (interior) transmission eigenfunctions associated with the Helmholtz system. It is shown in generic scenarios that there always exists a sequence of transmission eigenfunctions with the corresponding eigenvalues going to infinity such that those eigenfunctions are localized around the boundary of the domain. We provide a comprehensive and rigorous justification in the case within the radial geometry, whereas for the nonradial case, we conduct extensive numerical experiments to quantitatively verify the localizing behaviors. The discovery provides a new perspective on wave localization. As significant applications, we develop a novel inverse scattering scheme that can produce super-resolution imaging effects and propose a method of generating the so-called pseudo surface plasmon resonant (PSPR) modes with a potential sensing application.

ESTIMATES AND RIGIDITY FOR STABLE SOLUTIONS TO SOME NONLINEAR ELLIPTIC PROBLEMS

My thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabre, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincare sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.

Inward/outward energy theory of non-radial solutions to 3D semi-linear wave equation

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ∂t2u−Δu=−|u|p−1u in the 3-dimensional space with 3≤p<5. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if 3<p<5 and κ>5−p2, then the solution scatters as long as the initial data (u0,u1) satisfy∫R3(|x|κ+1)(12|∇u0|2+12|u1|2+1p+1|u0|p+1)dx<+∞. If p=3, we can also prove the scattering result if initial data (u0,u1) are contained in the critical Sobolev space and satisfy the inequality∫R3|x|(12|∇u0|2+12|u1|2+14|u0|p+1)dx<+∞. These assumptions on the decay rate of initial data as |x|→∞ are weaker than previously known scattering results in the non-radial case.

Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the L^2 -supercritical case

<p style='text-indent:20px;'>We consider the focusing <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle <inline-formula><tex-math id="M3">\begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. We construct a solution behaving asymptotically as a solitary wave on <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^3, $\end{document}</tex-math></inline-formula> for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

Adomain computation of radiative-convective bi-directional stretching flow of a magnetic non-Newtonian fluid in porous media with homogeneous–heterogeneous reactions

In the present communication, the laminar, incompressible, hydromagnetic flow of an electrically conducting non-Newtonian (Sisko) fluid over a bi-directional stretching sheet in a porous medium is studied theoretically. Thermal radiation flux, homogeneous–heterogeneous chemical reactions and convective wall heating are included in the model. The resultant nonlinear ordinary differential equations with transformed boundary conditions via similarity transformation are then solved with the semi-analytical Adomain Decomposition Method (ADM). Validation with earlier studies is included for the nonradiative case. Extensive visualization of velocity, temperature and species concentration distributions for various emerging parameters is included. Increasing the magnetic field and inverse permeability parameter is observed to decelerate both the primary and secondary velocity magnitudes whereas they increase temperatures in the regime. Increasing sheet stretching ratio weakly accelerates the primary flow throughout the boundary layer whereas it more dramatically accelerates the secondary flow near sheet surface. Temperature is consistently reduced with increasing stretching sheet ratio whereas it is strongly enhanced with greater radiative parameter. With greater Sisko non-Newtonian power-law index the primary velocity and temperature are decreased whereas the secondary velocity is increased. Increasing both homogenous and heterogeneous chemical reaction parameters is found to weakly and more strongly, respectively, deplete concentration magnitudes whereas greater Schmidt number enhances them.

The Lane-Emden equation with variable double-phase and multiple regime

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical, and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality.

Boundedness of stable solutions to nonlinear equations involving the p-Laplacian

We consider stable solutions to the equation −Δpu=f(u) in a smooth bounded domain Ω⊂Rn for a C1 nonlinearity f. Either in the radial case, or for some model nonlinearities f in a general domain, stable solutions are known to be bounded in the optimal dimension range n<p+4p/(p−1). In this article, under a new condition on n and p, we establish an L∞ a priori estimate for stable solutions which holds for every f∈C1. Our condition is optimal in the radial case for n≥3, whereas it is more restrictive in the nonradial case. This work improves the known results in the topic and gives a unified proof for the radial and the nonradial cases. The existence of an L∞ bound for stable solutions holding for all C1 nonlinearities when n<p+4p/(p−1) has been an open problem over the last twenty years. The forthcoming paper [11] by Cabré, Sanchón, and the author will solve it when p>2.

Instability of supersonic cold streams feeding galaxies – IV. Survival of radiatively cooling streams

ABSTRACT We study the effects of Kelvin–Helmholtz Instability (KHI) on the cold streams that feed massive haloes at high redshift, generalizing our earlier results to include the effects of radiative cooling and heating from a UV background, using analytic models and high resolution idealized simulations. We currently do not consider self-shielding, thermal conduction, or gravity. A key parameter in determining the fate of the streams is the ratio of the cooling time in the turbulent mixing layer which forms between the stream and the background following the onset of the instability, $t_{\rm cool,\, mix}$, to the time in which the mixing layer expands to the width of the stream in the non-radiative case, tshear. This can be converted into a critical stream radius, Rs, crit, such that $R_{\rm s}/R_{\rm s,crit}=t_{\rm shear}/t_{\rm cool,\, mix}$. If Rs &amp;lt; Rs, crit, the non-linear evolution proceeds similarly to the non-radiative case studied by Mandelker et al. If Rs &amp;gt; Rs,crit, which we find to almost always be the case for astrophysical cold streams, the stream is not disrupted by KHI. Rather, background mass cools and condenses on to the stream, and can increase the mass of cold gas by a factor of ∼3 within 10 stream sound crossing times. The mass entrainment induces thermal energy losses from the background and kinetic energy losses from the stream, which we model analytically. Roughly half of the dissipated energy is radiated away from gas with $T \lt 5\times 10^4\, {\rm K}$, suggesting much of it will be emitted in Ly α.

On a class of fully anisotropic elliptic equations

We consider a class of nonlinear elliptic equations driven by a fully anisotropic elliptic operator, and with a Carathéodory reaction. The elliptic operator depends on x and on the gradient of u, via the differential (with respect to u) of a general function belonging to a class of admissible functions. We present also several examples showing how our theorem incorporates the non-radial case, as well as that of functions with no polynomial growth.

Global center stable manifold for the defocusing energy critical wave equation with potential

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi, 0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It is constructed in a first step locally around any solution scattering to $\phi$, which might be very far away from $\phi$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends our previous result [18] to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from [3] to construct a local center stable manifold near any solution that scatters to $(\phi, 0)$. This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle [14], which is used to show that solutions that start off but near the local manifolds associated with $\phi$ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state $\phi$.