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 L...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.展开更多
The characteristic modifications are reported on the surface of polymeric waveguide film in the process of vol- ume-grating fabrication. The light from a mode-locked 76 MHz femtosecond laser with pulse duration of 200...The characteristic modifications are reported on the surface of polymeric waveguide film in the process of vol- ume-grating fabrication. The light from a mode-locked 76 MHz femtosecond laser with pulse duration of 200 fs and wavelength of 800 nm is focused normal to the surface of the sample. The surface morphology modifications are as- cribed to a fact that surface swelling occurs during the process. Periodic micro-structure is inscribed with increasing incident power. The laser-induced swelling threshold on the grating, which is higher than that of two-photon initiated photo-polymerization （TPIP） （8 mW）, is verified to be about 20 mW. It is feasible to enhance the surface smoothness of integrated optics devices for further encapsulation. The variation of modulation depth is studied for different values of incident power and scan spacing. Ablation accompanied with surface swelling appears when the power is higher. By ootimizing the laser carvinR oararneters, hizhly efficient grating devices can be fabricated.展开更多
This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spac...This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).展开更多
基金supported by the Central UniversitiesChina University of Geosciences(Wuhan)(CUGL180827)+1 种基金supported by the National Natural Science Foundation of China(11871218,12071298)supported by the National Natural Science Foundation of China(11771442)。
文摘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.
基金supported by the National Natural Science Foundation of China/the Research Grants Council of Hong Kong Joint Research Scheme a grant for NSFC/RGC(No.50218001)the National Science Foundation of China(No.50173015)
文摘The characteristic modifications are reported on the surface of polymeric waveguide film in the process of vol- ume-grating fabrication. The light from a mode-locked 76 MHz femtosecond laser with pulse duration of 200 fs and wavelength of 800 nm is focused normal to the surface of the sample. The surface morphology modifications are as- cribed to a fact that surface swelling occurs during the process. Periodic micro-structure is inscribed with increasing incident power. The laser-induced swelling threshold on the grating, which is higher than that of two-photon initiated photo-polymerization （TPIP） （8 mW）, is verified to be about 20 mW. It is feasible to enhance the surface smoothness of integrated optics devices for further encapsulation. The variation of modulation depth is studied for different values of incident power and scan spacing. Ablation accompanied with surface swelling appears when the power is higher. By ootimizing the laser carvinR oararneters, hizhly efficient grating devices can be fabricated.
基金supported by National Natural Science Foundation of China(Grant No.11731010)。
文摘This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).
