Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of ...Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the height of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.展开更多
The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in whic...The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.展开更多
.In this paper,we prove the global existence and uniqueness of so-lutions for the inhomogeneous Navier-Stokes equations with the initial data(ρ_(0),u_(0))∈L^(∞)×H^(s) _(0),s>1/2 and ||u_(0)||H^(s)_(0)≤ε0 ....In this paper,we prove the global existence and uniqueness of so-lutions for the inhomogeneous Navier-Stokes equations with the initial data(ρ_(0),u_(0))∈L^(∞)×H^(s) _(0),s>1/2 and ||u_(0)||H^(s)_(0)≤ε0 in bounded domain Ω■R^(3),in which the density is assumed to be nonnegative.The regularity of initial data is weaker than the previous(ρ_(0),u_(0))∈(W^(1)γ∩L^(∞)×H^(1)_(0) in [13] and(ρ_(0),u_(0))∈L^(∞)×H^(1)_(0) in[7],which constitutes a positive answer to the question raised by Danchin and Mucha in[7].The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach,and the continuity argu-ment and shift of integrability method are applied to complete our proof.展开更多
In either a periodic box Td or Rd （1 ≤ d ≤ 3）, we obtain global bounded solution of the relativistic Boltzmann equation near global relativistic Maxwellian, in terms of natural mass, energy conservation and the en...In either a periodic box Td or Rd （1 ≤ d ≤ 3）, we obtain global bounded solution of the relativistic Boltzmann equation near global relativistic Maxwellian, in terms of natural mass, energy conservation and the entropy inequality.展开更多
We study the periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation， Some theorems concerning the boundness, existence and uniqueness of solutions are proved，
文摘Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the height of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.
基金The second author(H.Cao)is supported by the National Natural Science Foundation of China(No.12001269)the Fundamental Research Funds for the Central Universities of China.
文摘.In this paper,we prove the global existence and uniqueness of so-lutions for the inhomogeneous Navier-Stokes equations with the initial data(ρ_(0),u_(0))∈L^(∞)×H^(s) _(0),s>1/2 and ||u_(0)||H^(s)_(0)≤ε0 in bounded domain Ω■R^(3),in which the density is assumed to be nonnegative.The regularity of initial data is weaker than the previous(ρ_(0),u_(0))∈(W^(1)γ∩L^(∞)×H^(1)_(0) in [13] and(ρ_(0),u_(0))∈L^(∞)×H^(1)_(0) in[7],which constitutes a positive answer to the question raised by Danchin and Mucha in[7].The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach,and the continuity argu-ment and shift of integrability method are applied to complete our proof.
文摘In either a periodic box Td or Rd （1 ≤ d ≤ 3）, we obtain global bounded solution of the relativistic Boltzmann equation near global relativistic Maxwellian, in terms of natural mass, energy conservation and the entropy inequality.
文摘We study the periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation， Some theorems concerning the boundness, existence and uniqueness of solutions are proved，
