The 4th-order spline wavelets an a bounded interval are constructed by the 4th-order truncated B-spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dim...The 4th-order spline wavelets an a bounded interval are constructed by the 4th-order truncated B-spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dimensions. Arty function on an interval will be expanded as the sum of finite items of the scaling functions and wavelets. It plays an important role for numerical analysis of partial differential equations, signal processes, and other similar problems.展开更多
Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t...Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t)=u+ and the initial data u(x,0)= u0(x, where um p u+ and f is a given function satisfying f'(u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When um < u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for um > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |um m u+| is small. Moreover, exponential decay rates are both given.展开更多
In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-...In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-1,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx≤c0|I|),andβq is optimal in the sense that u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx+∞),for anyβ>βq.Furthermore,when q is even,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ih(u)eβq|u(x)|q')dx≤+∞),for any function h:[0,∞→[0,∞)with lim t→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.展开更多
文摘The 4th-order spline wavelets an a bounded interval are constructed by the 4th-order truncated B-spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dimensions. Arty function on an interval will be expanded as the sum of finite items of the scaling functions and wavelets. It plays an important role for numerical analysis of partial differential equations, signal processes, and other similar problems.
基金Partially supported by the National Natural Sciences Foundation of China (No. 10101014), the Key Project of Natural Sciences Foundation of Beijing and Beijing Education Committee Foundation.Supported by the National Natural Science Foundation of China
文摘Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = ux,x on [0,1], with the boundary condition u(0,t)=um,u(1t)=u+ and the initial data u(x,0)= u0(x, where um p u+ and f is a given function satisfying f'(u>0 for u under consideration. By means of energy estimates method and under some more regular conditions on the initial data, both the global existence and the asymptotic behavior are obtained. When um < u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for um > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, which means that |um m u+| is small. Moreover, exponential decay rates are both given.
文摘In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-1,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx≤c0|I|),andβq is optimal in the sense that u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx+∞),for anyβ>βq.Furthermore,when q is even,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ih(u)eβq|u(x)|q')dx≤+∞),for any function h:[0,∞→[0,∞)with lim t→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.
