Remarks on the method of modulus of continuity and the modified dissipative Porous Media Equation

We employ Besov space techniques and the method of modulus of continuity to obtain the global well-posedness of the modified Porous Media Equation.     

Modulus of continuity of the coefficients and (non)quasianalytic solutions in the strictly hyperbolic Cauchy problem

In the strictly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultradifferentiable functions and functionals. We find well-posedness in nonquasianalytic classes assuming that the coefficients have modulus of continuity tω(1/t) such that . This condition is sharp because, in the case , we provide examples of Cauchy problems which are well-posed only in quasianalytic classes.     

Global well-posedness of the critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t_∂u+ux_∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where .     

Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.     

Global well-posedness of the hydrodynamic model for two-carrier plasmas

We prove results on the global well-posedness of the hydrodynamic model for two-carrier plasmas in whole space and periodic domain. We remove a technical condition which was first introduced by Alì and Jüngel [2] and developed in  and  to deal with the difficulty mainly arising from complicated coupling and cancellation between two carriers. The proofs depend on a result on continuity for compositions in Chemin–Lerner spaces and an elementary fact which indicates the connection between homogeneous and inhomogeneous Chemin–Lerner spaces.     

Initial–boundary value problems for continuity equations with BV coefficients

We establish well-posedness of initial–boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless of the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.     

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.     

On the Boussinesq/Full dispersion systems and Boussinesq/Boussinesq systems for internal waves

In this paper, we consider some asymptotic models for internal waves in the small amplitude/small amplitude regime, which were derived recently by Bona, Lannes and Saut. We first prove that the Boussinesq/Full dispersion systems and the Boussinesq/Boussinesq systems can be derived from the Full dispersion/Full dispersion systems. Then using a contraction-mapping argument and the energy method, we will prove that the derived systems that are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. In particular, we improve and extend some known results on the well-posedness of Boussinesq systems for surface waves.     

On general boundary-value problems for nonlinear elliptic equations of second order in a multiply connected plane domain

This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong.     

Некоторые точные нер авенства в теории при ближения функций

Exact inequalities are obtained that illuminate the interrelation between best polynomial approximations of functions, analytic in the disk and the modulus of continuity of the derivatives of the boundary values of these functions.For various classes of functions exact estimates are given for the derivative of a function by means of the modulus of continuity of this function and the modulus of continuity of its second derivative.As application, exact inequalities are deduced, analogous to the well-known Bernstein and Hardy inequalities.     

Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species

We prove the uniform Hölder continuity of solutions for two classes of singularly perturbed parabolic systems. These systems arise in Bose-Einstein condensates and in competing models in population dynamics. The proof relies upon the blow up technique and the monotonicity formulas by Almgren and Alt, Caffarelli, and Friedman.     

Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes

For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.     

The strong minimum principle for quasisuperminimizers of non-standard growth

We prove the strong minimum principle for non-negative quasisuperminimizers of the variable exponent Dirichlet energy integral under the assumption that the exponent has modulus of continuity slightly more general than Lipschitz. The proof is based on a new version of the weak Harnack estimate.     

Characterization of holomorphic functions in terms of their moduli

It was shown in (Boche, H. and Pohl, V., 2005, Spectral factorization in the disk algebra. Complex Variables. Theory and Applications, 50, 383–387.) that if the modulus |f| of a function is continuous in the closure of the unit disk, the function f itself needs not to be continuous there, in general. This article shows that if the modulus of continuity of a function is a weak regular majorant, the continuity of the modulus always implies the continuity of the function itself.     

Some remarks concerning the Fourier transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> (ℝ)

Two useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus.     

On a Problem of Gonska

This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.     

Best polynomial approximations and the widths of function classes in L 2

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived.     

On the low regularity of the fifth order Kadomtsev-Petviashvili I equation

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as , while α∈R. We introduce an interpolated energy space Es to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0<s?1. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate.