Class Of Large Data Research Articles

Global solutions to the Cauchy problem of BNSP equations in some classes of large data

Global solutions to the Cauchy problem of BNSP equations in some classes of large data

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Strong solutions of the equations for viscoelastic fluids in some classes of large data

We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. Our result mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preventing the formation of singularities in viscous flows. Moreover, for given initial velocity perturbation and zero initial deformation around the rest state, we find, as the elasticity coefficient or time goes to infinity, that(1)any straight line segment l0 consisted of fluid particles in the rest state, after being bent by a velocity perturbation, will turn into a straight line segment that is parallel to l0 and has the same length as l0.(2)the motion of the viscoelastic fluid can be approximated by a linear pressureless motion in Lagrangian coordinates, even when the initial velocity is large. Moreover, the above mentioned phenomena can also be found in the corresponding compressible fluid case.

Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity $\|\rho_0\|_{L^\infty}+\|b_0\|_{L^3}$ is suitably small and the viscosity coefficients satisfy $3\mu>\lambda$. Here, the initial velocity and initial temperature could be large. The assumption on the initial density do not exclude that the initial density may vanish in a subset of $\mathbb{R}^3$ and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu \cite{FY09} and Li et al. \cite{LXZ13}, where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension.

Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

On the Formation of Shock for Quasilinear Wave Equations with Weak Intensity Pulse

In this paper we continue to study the shock formation for the 3-dimensional quasilinear wave equation with $$G''(0)$$ being a non-zero constant. Since ( $$\star $$ ) admits global-in-time solution with small initial data, to present shock formation, we consider a class of large data. Moreover, no symmetric assumption is imposed on the data. Compared to our previous work (Miao and Yu in Invent Math 207(2):697–831, 2017), here we pose data on the hypersurface $$\{(t,x)|t=-\,r_{0}\}$$ instead of $$\{(t,x)|t=-\,2\}$$ , with $$r_{0}$$ being arbitrarily large. We prove an a priori energy estimate independent of $$r_{0}$$ . Therefore a complete description of the solution behavior as $$r_{0}\rightarrow \infty $$ is obtained. This allows us to relax the restriction on the profile of initial data which still guarantees shock formation. Since ( $$\star $$ ) can be viewed as a model equation for describing the propagation of electromagnetic waves in nonlinear dielectric, the result in this paper reveals the possibility to use wave pulse with weak intensity to form electromagnetic shocks in laboratory. A main new feature in the proof is that all estimates in the present paper do not depend on the parameter $$r_{0}$$ , which requires different methods to obtain energy estimates. As a byproduct, we prove the existence of semi-global-in-time solutions which lead to shock formation by showing that the limits of the initial energies exist as $$r_{0}\rightarrow \infty $$ . The proof combines the ideas in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2007) where the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established, and the short pulse method introduced in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012), where the formation of black holes in general relativity is proved.

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

We consider the Cauchy problem for the full compressible Navier--Stokes equations with vanishing of density at infinity in $\mathbb{R}^3$. Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data are more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of $\mathbb{R}^3$ and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first resul...

Nimble Protein Sequence Alignment in Grid (NPSAG)

In Bio-Informatics application, the analysis of protein sequence is a kind of computation driven science which has rapidly and quickly growing biological data. Also databases used in these applications are heterogeneous in nature and alignment of protein sequence using physical techniques is expensive, slow and results are not always guaranteed/accurate. So this application requires cross-platform, cost-effective and more computing power algorithm for sequence matching and searching a sequence in database. Grid is one of the most emerging technologies of cost effective computing paradigm for large class of data and compute intensive application which enables large-scale aggregation and sharing of computational data and other resources across institutional boundaries. We proposed the Grid architecture for searching of distributed, heterogeneous genomic databases which contained protein sequences to speed up the analysis of large scale sequence data and performed sequence alignment for residues match.