THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA

A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.

ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHR?DINGER EQUATION WITH LARGE INITIAL DATA

We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.

FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA L^p（P＜∞）

We study the existence problem for the equations of first order quasilinearequations in several inpendent variables with singular initial data Lp(P<∞). We the convergence of the Lp(P<∞) bounded approximating sequences generatedby the method of vanishing viscosity. The uniqueness of the generalized solutions whichcan be obtained by the method of vanishing viscosity is also obtained.

Large-time Behavior of Solutions for Parabolic Conservation Laws with Large Initial Data

In this paper,we study the large-time behavior of periodic solutions for parabolic conservation laws.There is no smallness assumption on the initial data.We firstly get the local existence of the solution by the iterative scheme,then we get the exponential decay estimates for the solution by energy method and maximum principle,and obtain the global solution in the same time.

CAUCHY PROBLEM FOR LINEARIZED SYSTEM OF TWO-DIMENSIONAL ISENTROPIC FLOW WITH AXISYMMETRICAL INITIAL DATA IN GAS DYNAMICS

The explicit solution to Cauchy problem for linearized system of two-dimensional isentropic flow with axisymmetrical initial data in gas dynamics is given.

ASYMPTOTIC HOMOGENIZATION IN A PARABOLIC SEMILINEAR PROBLEM WITH PERIODIC COEFFICIENTS AND INTEGRABLE INITIAL DATA

In this paper, we study the large time behavior of solutions of the parabolic semilinear equation δtu-div（a（x）△↓u） = -｜u｜^αu in （0,∞） × R^N, where α 〉 0 is constant and a∈ Cb^1（R^N） is a symmetric periodic matrix satisfying some ellipticity assumptions.Considering an integrable initial data u0 and α ∈ （2/N, 3/N）, we prove that the large time behavior of solutions is given by the solution U（t, x） of the homogenized linear problem δtU-div（a^h△↓U）=0,U（0） = C, where a^h is the homogenized matrix of a（x）, C is a positive constant and δ is the Dirac measure at 0.

ON THE CAUCHY PROBLEM OF THEKURAMOTO-SIVASHINSKY EQUATION WITH SINGULAR INITIAL DATA

In this paper, it is considered that the global existence, uniqueness and regularity results for the Cauchy problem of the well-known Kuramoto-Sivashinsky equation [GRAPHICS] only under the condition u(0)(x) is an element of L-2(R-N, R-n). Where u(t, x) = (u(1)(t, x), ..., u(n)(t, x))(T) is the unknown vector-valued function. Results show that for N < 6,.u(0)(x) is an element of L-2(R-N, R-n), the above Cauchy problem admits a unique global solution u(t, x) which belongs to C-infinity,C-infinity(R-N x (0, infinity)).

REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas.Based on the elaborate energy estimates,we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t)and W(t)introduced by Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].Our results also improve upon the results obtained in Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].

GLOBAL WELL-POSEDNESS FOR THE DENSITY-DEPENDENT INCOMPRESSIBLE MAGNETOHYDRODYNAMIC FLOWS IN BOUNDED DOMAINS

In this paper, we study the three-dimensional incompressible magnetohydrody-namic equations in a smooth bounded domains, in which the viscosity of the fluid and themagnetic diffusivity are concerned with density. The existence of global strong solutions isestablished in vacuum cases, provided the assumption that （｜ ｜μ（ρ0）｜｜Lp＋｜｜ v（P0）｜｜Lq＋｜｜b0｜｜L^3 ＋｜｜ρO｜｜L^∞） （p,q〉3） is small enough, there is not any smallness condition on thevelocity.

CAUCHY PROBLEM FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS

We consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient. For regular initial data, we show that the unique strong solution exits globally in time and converges to the equilibrium state time asymptotically. When initial density is piecewise regular with jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump discontinuity of the density decays exponentially and the piecewise regular solution tends to the equilibrium state as t →＋∞

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.

EXISTENCE AND UNIQUENESS OF THE GLOBAL L^(1) SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS

In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space Lloc1. The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system.The method used is Lagrangian representation, the essence of which is characteristic analysis.The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables.We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.

THE CAUCHY-KOVALEVSKAYA THEOREM-OLD AND NEW

The paper surveys interactions between complex and functional-analytic methods in the CauchyKovalevskaya theory. For instance, the behaviour of the derivative of a bounded holomorphic function led to abstract versions of the Cauchy-Kovalevskaya Theorem.Recent trends in the Cauchy-Kovalevskaya theory are based on the concept of associated differential operators. Since an evolution operator may posses several associated operators, initial data may be decomposed into components belonging to different associated spaces.This technique makes it also possible to solve ill-posed initial value problems.

Assimilation of Radar and Cloud-to-Ground Lightning Data Using WRF-3DVar Combined with the Physical Initialization Method——A Case Study of a Mesoscale Convective System

Radar data, which have incomparably high temporal and spatial resolution, and lightning data, which are great indicators of severe convection, have been used to improve the initial field and increase the accuracies of nowcasting and short-term forecasting. Physical initialization combined with the three-dimensional variational data assimilation method(PI3 DVarrh) is used in this study to assimilate two kinds of observation data simultaneously, in which radar data are dominant and lightning data are introduced as constraint conditions. In this way, the advantages of dual observations are adopted. To verify the effect of assimilating radar and lightning data using the PI3 DVarrh method, a severe convective activity that occurred on 5 June 2009 is utilized, and five assimilation experiments are designed based on the Weather Research and Forecasting(WRF) model. The assimilation of radar and lightning data results in moister conditions below cloud top, where severe convection occurs;thus, wet forecasts are generated in this study.The results show that the control experiment has poor prediction accuracy. Radar data assimilation using the PI3 DVarrh method improves the location prediction of reflectivity and precipitation, especially in the last 3-h prediction, although the reflectivity and precipitation are notably overestimated. The introduction of lightning data effectively thins the radar data, reduces the overestimates in radar data assimilation, and results in better spatial pattern and intensity predictions. The predicted graupel mixing ratio is closer to the distribution of the observed lightning,which can provide more accurate lightning warning information.

A Review on D 614G Mutation with Bangladesh Scenario

With the COVID-19 pandemic, disparities between the infection rate and death rate in different countries become a major concern. In some countries, lower mortality rate compared to others can be explained by better testing capacity and intensive care facilities. Complete SARS-CoV-2 genome sequences from different countries of the world are continually submitted to Global Initiative for Sharing All Influenza Data using Next Generation Sequencing method. A SARS-CoV-2 variant with a D 614G Mutation in the spike (S) protein has become the most dominant form in the global pandemic. There are a number of ongoing studies trying to relate this mutation with the infectivity, mortality, transmissibility of the virus and its impact on vaccine development. This review aims to accumulate the major findings from some of these studies and focus its future implication. Some studies suggested D 614G strain has increased binding capacity, it affects more cells at a faster rate, so has a high transmissibility. Patients infected with this strain were found with high viral load. But still now there is no such evidence that this strain produces more severe disease as well as increased mortality. The structural change of spike protein produced by D 614G mutation was minor and did not hamper the vaccine efficacy. Some studies showed antibodies produced against D614 strain can neutralize G614 strain and vice versa. Whenever a mutation occurs in spike protein there are always chances of affecting the infectivity, transmissibility, vaccine efficacy. Therefore, more studies are required to find out the overall effect of D 614G mutation.

Spatial propagation in an epidemic model with nonlocal diffusion:The influences of initial data and dispersals

This paper studies an epidemic model with nonlocal dispersals.We focus on the influences of initial data and nonlocal dispersals on its spatial propagation.Here,initial data stand for the spatial concentrations of the infectious agent and the infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies.Two types of initial data decaying to zero exponentially or faster are considered.For the first type,we show that spreading speeds are two constants whose signs change with the number of elements in some set.Moreover,we find an interesting phenomenon:the asymmetry of nonlocal dispersals can influence the propagating directions of the solutions and the stability of steady states.For the second type,we show that the spreading speed is decreasing with respect to the exponentially decaying rate of initial data,and further,its minimum value coincides with the spreading speed for the first type.In addition,we give some results about the nonexistence of traveling wave solutions and the monotone property of the solutions.Finally,some applications are presented to illustrate the theoretical results.

THE L1-ERROR ESTIMATES FOR A HAMILTONIAN-PRESERVING SCHEME FOR THE LIOUVILLE EQUATION WITH PIECEWISE CONSTANT POTENTIALS AND PERTURBED INITIAL DATA

We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l1-stability analysis in [46] and apply the Ll-error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is Ll-convergent when the initial data is given with a wide class of perturbation errors, and derive the Ll-error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l1-unstable.

INCOMPRESSIBLE LIMIT OF IDEAL MAGNETOHYDRODYNAMICS IN A DOMAIN WITH BOUNDARIES

We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is characteristic and the initial data is general.We first establish the uniform existence of classical solutions with respect to the Mach number.Then,we prove that the solutions converge to the solution of the incompressible MHD system.In particular,we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space.

Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution to the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system.In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called "effective viscous flux", which enables us to obtain the desired energy estimates and regularity. The technique of the "effective viscous flux" turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely(if not) used in the literature for chemotaxis models.

ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.