1,722,360 research outputs found
Use of Lawn Chemicals in the Twin Cities
Get PDFCreason, Jared R.; Runge, C. Ford. (1992). Use of Lawn Chemicals in the Twin Cities. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/219500
The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant
No full textOriginal manuscript January 24, 2012In this article, we study small perturbations of the family of Friedmann–Lemaître–Robertson–Walker cosmological background solutions to the 1 + 3 dimensional Euler–Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state p=c[2 over s]ρ, 0 < c[2 over s] < 1/3 , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0,∞) × T[superscript 3], are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents.National Science Foundation (U.S.). All-Institutes Postdoctoral Fellowship (Mathematical Sciences Research Institute (Berkeley, Calif.) Grant DMS-0441170
The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State
Get PDFAuthor's final manuscript January 10, 2012In this article, we study the 1 + 3-dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to R[superscript 3]. R 3 . We assume that the fluid verifies the equation of state p = c[2 over s]ρ, p = c s 2 ρ, where 0 ≤ c[subscript s] ≤ √1/3 0 ≤ c s ≤ 1/3 is the speed of sound. We also assume that the reciprocal of the scale factor associated with the expanding spacetime metric verifies a c[subscript s]−dependent time-integrability condition. Under these assumptions, we use the vector field energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions are globally future-stable under small perturbations of their initial conditions. The explicit solutions corresponding to each scale factor are analogs of the well-known spatially flat Friedmann–Lemaitre–Robertson–Walker family. Our nonlinear analysis, which exploits dissipative terms generated by the expansion, shows that the perturbed solutions exist for all future times and remain close to the explicit solutions. This work is an extension of previous results, which showed that an analogous stability result holds when the spacetime is exponentially expanding. In the case of the radiation equation of state p = (1/3)ρ, we also show that if the time-integrability condition for the reciprocal of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show the existence of an open family of initial data such that (i) it contains arbitrarily small smooth perturbations of the explicit solutions’ data and (ii) the corresponding perturbed solutions necessarily form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations when c[2 over s] =1/3, c s 2 = 1/3, which allows for a reduction to a well-known result of Christodoulou.National Science Foundation (U.S.) (Grant DMS-1162211)Solomon Buchsbaum Research FundNational Science Foundation (U.S.). All-Institutes Postdoctoral Fellowship (Mathematical Sciences Research Institute (Berkeley, Calif.) Grant DMS-0441170
Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations
Get PDFIn three spatial dimensions, we study the Cauchy problem for the wave equation −∂[superscript 2][subscript t]Ψ + (1+Ψ)[superscript P] ΔΨ=0 for P∈{1,2}. We exhibit a form of stable Tricomi-type degeneracy formation that has not previously been studied in more than one spatial dimension. Specifically, using only energy methods and ODE techniques, we exhibit an open set of data such that Ψ is initially near 0, while 1+Ψ vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this phenomenon. The solution remains regular in the following sense: there is a high-order L[superscript 2]-type energy, featuring degenerate weights only at the top-order, that remains bounded. When P = 1, we show that any C[superscript 1] extension of Ψ to the future of a point where 1 + Ψ = 0 must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the Lorentzian metric corresponding to the wave equation blows up at those points. Thus, our results show that curvature blowup does not always coincide with singularity formation in the solution variable. Similar phenomena occur when P = 2, where the vanishing of 1 + Ψ corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all values of Ψ.
The data are compactly supported and are allowed to be large or small as measured by an unweighted Sobolev norm. However, we assume that initially the spatial derivatives of Ψ are nonlinearly small relative to |∂[subscript t]Ψ|, which allows us to treat the equation as a perturbation of the ODE (d[superscript 2]/dt[superscript 2])Ψ = 0. We show that for appropriate data, ∂tΨ remains quantitatively negative, which simultaneously drives the degeneracy formation and yields a favorable spacetime integral in the energy estimates that is crucial for controlling some top-order error terms. Our result complements those of Alinhac and Lindblad, who showed that if the data are small as measured by a Sobolev norm with radial weights, then the solution is global. Keywords: degenerate hyperbolic; strictly hyperbolic; Tricomi equation; weakly hyperbolicNational Science Foundation (U.S.) (Grant DMS-1162211)National Science Foundation (U.S.) (Grant DMS-1454419
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
On the questions of local and global well-posedness for the hyperbolic PDEs occurring in some relativistic theories of gravity and electromagnetism
No full textThe two hyperbolic systems of PDEs we consider in this work are the source-free Maxwell-Born-Infeld (MBI) field equations and the Euler-Nordstr??m system for gravitationally self-interacting fluids. The former system plays a central role in Kiessling's recently proposed self-consistent model of classical
electrodynamics with point charges, a model that does not suffer from the infinities found in the classical Maxwell-Maxwell model with point charges. The latter system is a scalar gravity caricature of the incredibly more complex Euler-Einstein system. The primary original contributions of the thesis can be summarized as follows:
1) We give a sharp non-local criterion for the formation of singularities in plane-symmetric solutions to the source-free MBI field equations. We also use a domain of dependence argument to show that 3-d initial data agreeing with certain plane-symmetric data on a large enough ball lead to solutions that form singularities in finite time. This work is an extension of a theorem of Brenier, who studied singularity formation in periodic plane-symmetric solutions.
2) We prove well-posedness for the Euler-Nordstr??m system with a cosmological constant k (EN_k) for initial data that are an H^N perturbation (not necessarily small) of a uniform, quiet fluid, for N [greater than]= 3. The method of proof relies on the framework of energy currents that has been recently developed by Christodoulou. We turn to this method out of necessity: two common frameworks for showing well-posedness in H^N, namely symmetric hyperbolicity and strict hyperbolicity, do not apply to the EN_k system, while Christodoulou's techniques apply to all hyperbolic systems derivable from a Lagrangian, of which the EN_k system is an example.
3) We insert the speed of light c as a parameter into the EN_k system (and designate the family of systems EN_k^c) in order to study the non-relativistic limit c to infinity. Taking the formal limit in the equations gives the Euler-Poisson system with a cosmological constant (EP_k). Using energy currents, we prove that for fixed initial data, as c goes to infinity, the solutions to the EN_k^c system converge uniformly on a spacetime slab [0,T] x R^3 to the solution of the EP_k system.Ph.D.Includes bibliographical references (p. 140-143)
The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant
No full textWe study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 3. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state [mathematical equation]. The dust-Einstein system is the case c[subscript s] = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases [mathematical equation] which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions
Shock Formation for 2D Quasilinear Wave Systems Featuring Multiple Speeds: Blowup for the Fastest Wave, with Non-trivial Interactions up to the Singularity
No full textWe prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formation results for other related systems with many unknowns and multiple speeds, in various solution regimes, and in higher spatial dimensions. However, a fundamental aspect of our framework is that it applies only to solutions in which the “fastest wave” forms a shock while the remaining solution variables do not, even though they can be non-zero at the fastest wave’s singularity. Our approach is based on an extended version of the geometric vectorfield method developed by D. Christodoulou in his study of shock formation for scalar wave equations, as well as the framework that we developed in our joint work with J. Luk, in which we proved a shock formation result for a quasilinear wave-transport system featuring a single wave operator. A key new difficulty that we encounter is that the geometric vectorfields that we use to commute the equations are, by necessity, adapted to the wave operator of the (shock-forming) fast wave and therefore exhibit very poor commutation properties with the slow wave operator, much worse than their commutation properties with a transport operator. In fact, commuting the vectorfields all the way through the slow wave operator would create uncontrollable error terms. To overcome this difficulty, we rely on a first-order reformulation of the slow wave equation, which, though somewhat limiting in the precision it affords, allows us to avoid uncontrollable commutator terms. Keywords: Characteristics, Eikonal equation, Eikonal function, Genuinely nonlinear strictly hyperbolic systems, Null condition, Null hypersurface, Singularity formation, Strong null condition, Vectorfield method, Wave breakingNational Science Foundation (U.S.) (Grant DMS-1162211)National Science Foundation (U.S.) (CAREER Grant DMS-1454419)Alfred P. Sloan FoundationSolomon Buchsbaum AT&T Research Fun
The nonlinear stability of the trivial solution to the Maxwell-Born-Infeld system
No full textIn this article, we use an electromagnetic gauge-free framework to establish the existence of small-data global solutions to the Maxwell-Born-Infeld (MBI) system on the Minkowski spacetime background in 1+3 dimensions. Because the nonlinearities in the system have a special null structure, we are also able to show that these solutions decay at least as fast as solutions to the linear Maxwell-Maxwell system. In addition, we show that on any Lorentzian manifold, the MBI system is hyperbolic in the interior of the field-strength regime in which its Lagrangian is real-valued.National Science Foundation (U.S.) (Grant DMS-0406951)National Science Foundation (U.S.) (Grant DMS-0807705
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