Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrodinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

/pdf/endpoint-strichartz-estimates-f9o0p7f76l.pdf

Endpoint Strichartz estimates

Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

/pdf/nonlinear-dispersive-equations-local-and-global-analysis-527whycdsg.pdf

Nonlinear dispersive equations : local and global analysis

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.

/pdf/remarks-on-the-breakdown-of-smooth-solutions-for-the-3-d-19bvo5v2sw.pdf

Remarks on the breakdown of smooth solutions for the 3- D Euler equations

WAVE BREAKING FOR NONLINEAR NONLOCAL SHALLOW WATER EQUATIONS 233 THEOREM 2.1. Let T>O and vE C 1 ([0, T); H 2(R)). Then for every t~ [0, T) there exists at least one point ~(t)ER with ,~(t) := in~ Ivy(t, x)] = ~ ( t , ~(t)), and the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t,~(t)) a.e. on (O,T). Proof. Let c>0 stand for a generic constant. Fix te[0, T) and define m(t):=infxcR[v~(t,x)]. If m(t))O we have that v(t , . ) is nondecreasing on R and therefore v(t,. ) 0 (recall v(t,. )cL2(R)) , so that we may assume re(t)<0. Since vx(t,. ) c H I ( R ) we see that limlxl~ ~ Vx(t, x)=0 so that there exists at least a ~(t) e R with re(t) =v~(t, ~(t)). Let now s, tC [0, T) be fixed. If re(t) <~rn(s) we have 0 < re(s) .~(t) = i n f [~x (~, x ) ] ~ ( t , ~(t)) <. ~=(~, ~(t)) -~x(t, ~(t)), and by the Sobolev embedding HI(R) C L ~ ( R ) we conclude that Im(8)-.~(t)l ~< Ivx(t)-v~(s)lL~(~) < c Iv~(t)-v~(8)l.l(R). Hence the mean-value theorem for functions with values in Banach spaces-Hi(R) in the present case--yields (see [12]) jm(t)-m(s)l<~clt-s j m a x [IVt~(T)JHI(R)], t, se[O,T). O~T~max{s,t} Since vt~cC([O,T), Hi(R)) , we see that m is locally Lipschitz on [0, T) and therefore Rademacher's theorem (cf. [14]) implies that m is almost everywhere differentiable on (0,T). Fix tC(0, T). We have that v~(t+h)-vx(t)h vt~(t) Hl(R) ---~0 as h--*O, and therefore vx( t+h ,y ) -vx ( t , y ) sup vtx(t,y) --~0 as h---~O, (2.1) ycl~ h in view of the continuous embedding H 1 ( R ) c L ~ (R). 234 A. C O N S T A N T I N AND J. E S C H E R By the definition of m, m(t+h) = v~(t+h, ((t+h)) <. v~(t+h, ((t)). Consequently, given h>0 , we obtain m( t+h) -m( t ) <<. h Letting h--~O + and using (2.1), we find lim sup m(t+h) -m( t ) h~_~0 + h

/pdf/wave-breaking-for-nonlinear-nonlocal-shallow-water-equations-2y25ekrd18.pdf

Wave breaking for nonlinear nonlocal shallow water equations

The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which a rigorous derivation of the laws of gravitation proposed by Bondi can be given. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.

The Global Nonlinear Stability of the Minkowski Space.

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

On etudie le comportement asymptotique des solutions de □u=0 ou □=∂ t 2 −∂ 1 2 ...−∂ n 2 pour des conditions initiales u=0, u t =g(x) en t=0, avec g reguliere a support compact dans R n

Uniform decay estimates and the lorentz invariance of the classical wave equation

Incompressible fluid: are the fluids with constant density. They could be liquids and gases. Although there is no such thing in reality as an incompressible fluid, we use this term where the change in density with pressure is so small as to be negligible. This is usually the case with liquids. We may also consider gases to be incompressible when the pressure variation is small compared with the absolute pressure. In problems involving water hammer we must consider the compressibility of the liquid. The flow of air in a ventilating system is a case where we may treat a gas as incompressible, for the pressure variation is so small that the change in density is of no importance. But for a gas or steam flowing at high velocity through a long pipeline, the drop in pressure may be so great that we cannot ignore the change in density. For an airplane flying at speeds below 250 mph (100 m/s), we may consider the air to be of constant density. But as an object moving through the air approaches the velocity of sound, which is of the order of 760 mph (1200 km/h) depending on temperature, the pressure and density of the air adjacent to the body become materially different from those of the air at some distance away, and we must then treat the air as a compressible fluid.

Sergiu Klainerman

Papers

The Global Nonlinear Stability of the Minkowski Space.

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Uniform decay estimates and the lorentz invariance of the classical wave equation

Compressible and incompressible fluids

Space-time estimates for null forms and the local existence theorem