The Experts below are selected from a list of 258 Experts worldwide ranked by ideXlab platform
Daniel Tataru - One of the best experts on this subject based on the ideXlab platform.
-
The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions
Pure and Applied Analysis, 2020Co-Authors: Daniel TataruAbstract:This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider Topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge. The results in this paper are used in the subsequent papers arXiv:1709.08604, arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the trivial Topological Class, and the dichotomy theorem in arbitrary Topological Classes.
-
the hyperbolic yang mills equation for connections in an arbitrary Topological Class
Communications in Mathematical Physics, 2019Co-Authors: Daniel TataruAbstract:This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling versus Scattering Dichotomy for the energy critical hyperbolic Yang–Mills equation in the (4 + 1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang–Mills equation in an arbitrary Topological Class at an optimal regularity. We generalize the standard notion of a Topological Class of connections on $${\mathbb{R}^{d}}$$ , defined via a pullback to the one-point compactification $${\mathbb{S}^{d} = \mathbb{R}^{d} \cup \{\infty}\}$$ , to rough connections with curvature in the critical space $${L^{\frac{d}{2}}(\mathbb{R}^{d})}$$ . Moreover, we provide excision and extension techniques for the Yang–Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang–Mills initial data sets. Combined with the results in the previous paper (Oh and Tataru in The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, 2017. arXiv:1709.09332 ), we obtain local well-posedness of the hyperbolic Yang–Mills equation on $${\mathbb{R}^{1+d}}$$ $${(d \geq 4)}$$ in an arbitrary Topological Class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d = 3, our techniques provide an alternative proof of the Classical finite energy global well-posedness theorem of Klainerman–Machedon (Ann. Math. (2) 142(1):39–119, 1995. https://doi.org/10.2307/2118611 ), while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao (J. Differ. Equ. 189(2):366–382, 2003. https://doi.org/10.1016/S0022-0396(02)00177-8 ). Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.
-
The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class
Communications in Mathematical Physics, 2019Co-Authors: Daniel TataruAbstract:This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling versus Scattering Dichotomy for the energy critical hyperbolic Yang–Mills equation in the (4 + 1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang–Mills equation in an arbitrary Topological Class at an optimal regularity. We generalize the standard notion of a Topological Class of connections on $${\mathbb{R}^{d}}$$ R d , defined via a pullback to the one-point compactification $${\mathbb{S}^{d} = \mathbb{R}^{d} \cup \{\infty}\}$$ S d = R d ∪ { ∞ } , to rough connections with curvature in the critical space $${L^{\frac{d}{2}}(\mathbb{R}^{d})}$$ L d 2 ( R d ) . Moreover, we provide excision and extension techniques for the Yang–Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang–Mills initial data sets. Combined with the results in the previous paper (Oh and Tataru in The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, 2017 . arXiv:1709.09332 ), we obtain local well-posedness of the hyperbolic Yang–Mills equation on $${\mathbb{R}^{1+d}}$$ R 1 + d $${(d \geq 4)}$$ ( d ≥ 4 ) in an arbitrary Topological Class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d = 3, our techniques provide an alternative proof of the Classical finite energy global well-posedness theorem of Klainerman–Machedon (Ann. Math. (2) 142(1):39–119, 1995 . https://doi.org/10.2307/2118611 ), while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao (J. Differ. Equ. 189(2):366–382, 2003 . https://doi.org/10.1016/S0022-0396(02)00177-8 ). Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.
-
The hyperbolic Yang--Mills equation for connections in an arbitrary Topological Class
Communications in Mathematical Physics, 2018Co-Authors: Daniel TataruAbstract:This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the $(4+1)$-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang--Mills equation in an arbitrary Topological Class at an optimal regularity. We generalize the standard notion of a Topological Class of connections on $\mathbb R^{d}$, defined via a pullback to the one-point compactification $\mathbb S^{d} = \mathbb R^{d} \cup \{\infty\}$, to rough connections with curvature in the critical space $L^{\frac{d}{2}}(\mathbb R^{d})$. Moreover, we provide excision and extension techniques for the Yang--Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang--Mills initial data sets. Combined with the results in the previous paper \cite{OTYM2}, we obtain local well-posedness of the hyperbolic Yang--Mills equation on $\mathbb R^{1+d}$ $(d \geq 4)$ in an arbitrary Topological Class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case $d = 3$, our techniques provide an alternative proof of the Classical finite energy global well-posedness theorem of Klainerman--Machedon \cite{KlMa2}, while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao \cite{TaoYM}. Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.
-
The threshold conjecture for the energy critical hyperbolic Yang--Mills equation
arXiv: Analysis of PDEs, 2017Co-Authors: Daniel TataruAbstract:Author(s): Oh, Sung-Jin; Tataru, Daniel | Abstract: This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical $4+1$ dimensional hyperbolic Yang--Mills equation. The Threshold Theorem asserts that Topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary Topological Class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary Topological Class at optimal regularity in the third paper, here we perform a blow-up analysis which shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.
Tarasankar Debroy - One of the best experts on this subject based on the ideXlab platform.
-
Grain topology in Ti 6Al 4V welds—Monte Carlo simulation and experiments
Journal of Physics D: Applied Physics, 2004Co-Authors: Sushil Mishra, Tarasankar DebroyAbstract:The importance of Topological features of grains in the evolution of grain structure is well recognized in isothermal systems. However, during fusion welding, strong spatial gradients of temperature exist in the heat-affected zone (HAZ), and this region undergoes rapid heating and cooling. The effects of spatial and temporal variations of temperature on the Topological Class distribution, relationship between size and topology of grains and the interdependence between grain topology and its neighbours are not known. Topological features of grains in the HAZ of Ti–6Al–4V alloy welds were measured for various heat inputs in the range 0.55–4.33 MJ m−1. The Topological Class distributions were also calculated using a three-dimensional Monte Carlo model utilizing thermal cycles computed from a well tested numerical heat transfer and fluid flow model. The computed results showed that the Topological Class distributions were unaffected by the spatial and temporal variations of temperature. Experimental investigations of a few sections confirmed the simulation results. The average grain size for each edge Class varied linearly with the edge Class number. The local Topological environment, i.e. the average number of sides of neighbours, nn, varied linearly with the inverse of the number of sides of grains, 1/nr, at a given location in the HAZ. Locations with the same Topological environment showed the same grain size, indicating the significant influence of grain topology on grain growth in the HAZ.
-
Measurements and Monte Carlo simulation of grain growth in the heat-affected zone of Ti–6Al–4V welds
Acta Materialia, 2004Co-Authors: Sushil Mishra, Tarasankar DebroyAbstract:Abstract Grain size and Topological Class distributions in the heat-affected zone (HAZ) of gas tungsten arc welded Ti–6Al–4V alloy were measured for various heat inputs in the range of 0.55–4.33 MJ m −1 . The evolution of grain structure and Topological Class distributions were also calculated using a three-dimensional Monte Carlo model utilizing thermal cycles computed from a well tested numerical heat transfer and fluid flow model. Both the experimental data and the calculated results showed that the average prior-β grain size near the fusion plane was about four to twelve times larger than the average grain size in the base plate, depending on the heat input. At locations equidistant from the fusion plane, the grains were larger in the mid-section vertical symmetry plane as compared to those at the top surface due to local variations of the thermal cycles. The normalized grain size distributions were unaffected by the local differences in the thermal cycles. It is demonstrated that the presence of a spatial gradient of temperature in the HAZ significantly impeded grain growth due to thermal pinning effect. Furthermore, the steep temperature gradients near the fusion plane did not introduce any significant texture in the grains. Both the experimental data and the calculated results indicated that the grains in the HAZ of the Ti–6Al–4V alloy were significantly smaller than the grains in the commercially pure titanium for identical welding conditions.
-
Three-dimensional monte carlo simulation of grain growth in zone-refined iron
Metallurgical and Materials Transactions B, 2001Co-Authors: S. Sista, Tarasankar DebroyAbstract:The evolution of the grain structure and Topological-Class distributions in zone-refined iron were modeled using a three-dimensional (3-D) Monte Carlo (MC) model. The effect of grain size on Topological features was examined. The relationship between the Topological features of grains and the geometry of their surrounding grains was studied. In particular, the average number of sides of the grains was related to the average number of sides of their neighbors. Both the computed grain-size distribution and the Topological-Class distribution were found to be invariant with time. A linear relationship existed between the average grain size and the average number of sides of grains. The number of sides of grains was inversely proportional to the average number of sides of the neighboring grains. The computed average grain size, grain-size distribution, and Topological-Class distribution agreed well with the corresponding independent experimental data. The results indicate significant promise for understanding grain growth and Topological features using 3-D MC simulation.
Sushil Mishra - One of the best experts on this subject based on the ideXlab platform.
-
Grain topology in Ti 6Al 4V welds—Monte Carlo simulation and experiments
Journal of Physics D: Applied Physics, 2004Co-Authors: Sushil Mishra, Tarasankar DebroyAbstract:The importance of Topological features of grains in the evolution of grain structure is well recognized in isothermal systems. However, during fusion welding, strong spatial gradients of temperature exist in the heat-affected zone (HAZ), and this region undergoes rapid heating and cooling. The effects of spatial and temporal variations of temperature on the Topological Class distribution, relationship between size and topology of grains and the interdependence between grain topology and its neighbours are not known. Topological features of grains in the HAZ of Ti–6Al–4V alloy welds were measured for various heat inputs in the range 0.55–4.33 MJ m−1. The Topological Class distributions were also calculated using a three-dimensional Monte Carlo model utilizing thermal cycles computed from a well tested numerical heat transfer and fluid flow model. The computed results showed that the Topological Class distributions were unaffected by the spatial and temporal variations of temperature. Experimental investigations of a few sections confirmed the simulation results. The average grain size for each edge Class varied linearly with the edge Class number. The local Topological environment, i.e. the average number of sides of neighbours, nn, varied linearly with the inverse of the number of sides of grains, 1/nr, at a given location in the HAZ. Locations with the same Topological environment showed the same grain size, indicating the significant influence of grain topology on grain growth in the HAZ.
-
Measurements and Monte Carlo simulation of grain growth in the heat-affected zone of Ti–6Al–4V welds
Acta Materialia, 2004Co-Authors: Sushil Mishra, Tarasankar DebroyAbstract:Abstract Grain size and Topological Class distributions in the heat-affected zone (HAZ) of gas tungsten arc welded Ti–6Al–4V alloy were measured for various heat inputs in the range of 0.55–4.33 MJ m −1 . The evolution of grain structure and Topological Class distributions were also calculated using a three-dimensional Monte Carlo model utilizing thermal cycles computed from a well tested numerical heat transfer and fluid flow model. Both the experimental data and the calculated results showed that the average prior-β grain size near the fusion plane was about four to twelve times larger than the average grain size in the base plate, depending on the heat input. At locations equidistant from the fusion plane, the grains were larger in the mid-section vertical symmetry plane as compared to those at the top surface due to local variations of the thermal cycles. The normalized grain size distributions were unaffected by the local differences in the thermal cycles. It is demonstrated that the presence of a spatial gradient of temperature in the HAZ significantly impeded grain growth due to thermal pinning effect. Furthermore, the steep temperature gradients near the fusion plane did not introduce any significant texture in the grains. Both the experimental data and the calculated results indicated that the grains in the HAZ of the Ti–6Al–4V alloy were significantly smaller than the grains in the commercially pure titanium for identical welding conditions.
Rahul Roy - One of the best experts on this subject based on the ideXlab platform.
-
Z 2 Classification of quantum spin Hall systems: An approach using time-reversal invariance
Physical Review B, 2009Co-Authors: Rahul RoyAbstract:We study the phases of Bloch insulators with time-reversal symmetry on the basis of the homotopy of the ground-state wave functions in momentum space and find that there are two Topological Classes characterized by a ${Z}_{2}$ invariant. The results are in agreement with a recent study based on counting the zeroes of a certain Pfaffian function related to the ground-state wave function. It is shown that there is a link between the formulation of the Topological invariant presented here and the number of robust edge states. A formula is also provided which greatly simplifies the computation of the invariant in a large number of cases. The present study provides guidance for the search of systems which belong to the nontrivial Topological Class and also establishes a link between the quantum spin Hall effect and the integer quantum Hall effect.
-
Topological superfluids with time reversal symmetry
arXiv: Mesoscale and Nanoscale Physics, 2008Co-Authors: Rahul RoyAbstract:It is shown that superfluids in two and three dimensions which have time reversal invariant ground states have phases which are distinguished by a Topological invariant. Further, it is shown that the B-phase of $^3$ He is a superfluid in the non-trivial Topological Class. Superfluids in the non-trivial Topological Class are shown to have gapless edge states and support various kinds of vortices with zero energy modes localized in their cores. Some of these vortices have non-abelian statistics.
Max Lein - One of the best experts on this subject based on the ideXlab platform.
-
Krein-Schrödinger formalism of bosonic Bogoliubov–de Gennes and certain Classical systems and their Topological Classification
Physical Review B, 2019Co-Authors: Max Lein, Koji SatoAbstract:To understand recent works on Classical and quantum spin equations and their Topological Classification, we develop a unified mathematical framework for bosonic Bogoliubov--de Gennes (BdG) systems and associated Classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG Hamiltonian. Because here the generator of dynamics, the analog of the Hamiltonian, is para-Hermitian (also known as pseudo- or Krein-Hermitian) but not Hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the Classical equations can be expressed as a ``Schr\"odinger equation'' with a Hermitian Hamiltonian. We then apply the Cartan-Altland-Zirnbauer Classification scheme: To properly understand what Topological Class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since Classical waves are necessarily real-valued. Because of this distinction, only commuting symmetries enter in the Topological Classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of Class A, the same Topological Class as quantum Hamiltonians describing the integer quantum Hall effect. Consequently, the magnonic edge modes first predicted by Shindou et al. [Phys. Rev. B 87, 174427 (2013)] are indeed analogs of the quantum Hall effect, and their net number is Topologically protected.
-
The Krein-Schr\"odinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification
arXiv: Other Condensed Matter, 2019Co-Authors: Max Lein, Koji SatoAbstract:To understand recent works on Classical and quantum spin equations and their Topological Classification, we develop a unified mathematical framework for bosonic BdG systems and associated Classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG hamiltonian. Because here the generator of dynamics, the analog of the hamiltonian, is para-, aka Krein-, hermitian but not hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the Classical equations can be expressed as a Schrodinger equation with a hermitian hamiltonian. We then proceed to apply the Cartan-Altland-Zirnbauer Classification scheme: to properly understand what Topological Class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since Classical waves are necessarily real-valued. Because of this distinction only commuting symmetries enter in the Topological Classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of Class A, the same Topological Class as quantum hamiltonians describing the Integer Quantum Hall Effect. Consequently, the magnonic edge modes first predicted by Shindou et al. are indeed analogs of the Quantum Hall Effect, and their net number is Topologically protected.
-
Derivation of Ray Optics Equations in Photonic Crystals via a SemiClassical Limit
Annales Henri Poincaré, 2017Co-Authors: Giuseppe Nittis, Max LeinAbstract:In this work, we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiClassical technique. We implement this scheme for periodic light conductors, photonic crystals , thereby making the quantum-light analogy between semiClassics for the Bloch electron and ray optics in photonic crystals rigorous. One major conceptual difference between the two theories, though, is that electromagnetic fields are real, and hence, we need to add one step in the derivation to reduce it to a single-band problem. Our main results, Theorem 3.7 and Corollary 3.9, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all subleading-order terms, some of which are also new to the physics literature. The ray optics limit we prove applies to photonic crystals of any Topological Class .