A large class of stochastic differential games for several players is considered in this paper. The class includes Nash differential games as well as Stackelberg differential games. A mix is possible. The existence of feedback strategies under general conditions is proved. The limitations concern the functionals in which the state and the controls appear separately. This is also true for the state equations. The controls appear in a quadratic form for the payoff and linearly in the state equation. The most serious restriction is the dimension of the state equation, which cannot exceed 2. The reason comes from PDE (partial differential equations) techniques used in studying the system of Bellman equations obtained by Dynamic Programming arguments. In the authors’ previous work in 2002, there is not such a restriction, but there are serious restrictions on the structure of the Hamiltonians, which are violated in the applications dealt with in this article.
For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition, in which all the characteristics, except a weakly linearly degenerate one, are involved in the dissipation, the global existence of H 2 classical solution to the Cauchy problem with small initial data is obtained.
The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws. Firstly, a kinetic formulation of entropy solutions is established. Secondly, by using the kinetic formulation and kinetic techniques, the uniqueness of entropy solutions is obtained. Finally, the parabolic approximation is studied and an error estimate of order $\eta ^{\tfrac{1}{3}}$ between the entropy solution and the viscous approximate solutions is established by using kinetic techniques, where η is the size of artificial viscosity.
This paper contains a study of propagation of singular travelling waves u(x, t) for conservation laws u t +[ϕ(u)] x = ψ(u), where ϕ, ψ are entire functions taking real values on the real axis. Conditions for the propagation of wave profiles β + mδ and β + mδ′ are presented (β is a real continuous function, m ≠ 0 is a real number and δ′ is the derivative of the Dirac measure δ). These results are obtained with a consistent concept of solution based on our theory of distributional products. Burgers equation $u_t + \left( {\tfrac{{u^2 }}{2}} \right)_x = 0$, the diffusionless Burgers-Fischer equation $u_t + a\left( {\tfrac{{u^2 }}{2}} \right)_x = ru\left( {1 - \tfrac{u}{k}} \right)$ with a, r, k being positive numbers, Leveque and Yee equation $u_t + u_x = uu\left( {1 - u} \right)\left( {u - \tfrac{1}{2}} \right)$ with μ ≠ 0, and some other examples are studied within such a setting. A “tool box” survey of the distributional products is also included for the sake of completeness.
The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible, isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investigated in this paper. In a paper (Comm. Pure Appl. Math., 46, 1993, 621–665) by Z. P. Xin, the author constructed a sequence of solutions to one-dimensional Navier-Stokes isentropic equations converging to the rarefaction wave as the viscosity tends to zero. Furthermore, he obtained that the convergence rate is $\varepsilon ^{\tfrac{1}{4}} \left| {\ln \varepsilon } \right|$. In this paper, Xin’s convergence rate is improved to $\varepsilon ^{\tfrac{1}{3}} \left| {\ln \varepsilon } \right|^2$ by different scaling arguments. The new scaling has various applications in related problems.
The boundary controllability of the fourth order Schrödinger equation in a bounded domain is studied. By means of an L 2-Neumann boundary control, the authors prove that the solution is exactly controllable in H −2(Ω) for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.
The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large. The exponential decay estimates of the solutions are obtained for the power of Laplacian α ∈ [1/2, 1).
In this work, by virtue of the properties of weakly almost periodic points of a dynamical system (X, T) with at least two points, the authors prove that, if the measure center M(T) of T is the whole space, that is, M(T) = X, then the following statements are equivalent:
| 1. | (X, T) is ergodic mixing |
| 2. | (X, T) is topologically double ergodic |
| 3. | (X, T) is weak mixing |
| 4. | (X, T) is extremely scattering |
| 5. | (X, T) is strong scattering |
| 6. | (X × X, T × T) is strong scattering |
| 7. | (X × X, T × T) is extremely scattering |
| 8. | For any subset S of ℕ with upper density 1, there is a c-dense F σ-chaotic set with respect to S. |
As an application, the authors show that, for the sub-shift σ A of finite type determined by a k × k-(0, 1) matrix A, σ A is strong mixing if and only if σ A is totally transitive.
The asymptotic distribution of the change-point estimator in a jump changepoint model is considered. For the jump change-point model X i = a + θI{[nτ 0] < i ≤ n} + ɛ i, where ɛ i (i = 1, ..., n) are independent identically distributed random variables with Eɛ i = 0 and Var(ɛ i) < ∞, with the help of the slip window method, the asymptotic distribution of the jump change-point estimator $\hat \tau$ is studied under the condition of the local alternative hypothesis.
This paper establishes the global existence of classical solution to the system of homogeneous, isotropic hyperelasticity with time-independent external force, provided that the nonlinear term obeys a type of null condition. The authors first prove the existence and uniqueness of the stationary solution. Then they show that the solution to the dynamical system converges to the stationary solution as time goes to infinity.
The paper is concerned with the system modeling the compressible hydrodynamic flow of liquid crystals with radially symmetric initial data and non-negative initial density in dimension N (N ≥ 2). The authors obtain the existence of global radially symmetric strong solutions in a bounded or unbounded annular domain for any γ > 1.