Volume 15, Issue 3

This paper considers an established model of a parabolic-hyperbolic coupled system of two PDEs, which arises when an elastic structure is immersed in a fluid. Coupling occurs at the interface between the two media. Semigroup well-posedness on the space of finite energy for {𝑤, 𝑤 𝑡 , 𝑢} was established in [Contemp. Math. 440: 15–54, 2007]. Here, [𝑤, 𝑤 𝑡 ] are the displacement and the velocity of the structure, while 𝑢 is the velocity of the fluid. The domain D ( A ) of the generator A does not carry any smoothing in the 𝑤-variable (its resolvent 𝑅(λ, A ) is not compact on this component space). This raises the issue of higher regularity of solutions. This paper then shows that the mechanical displacement, fluid velocity, and pressure terms do enjoy a greater regularity if, in addition to the I.C. {𝑤 0 , 𝑤 1 , 𝑢 0 } ∈ D ( A ), one also has 𝑤 0 in (𝐻 2 (Ω 𝑠 )) 𝑑 .

We present here impulse control theory in discrete time. We show the connection between quasi-variational inequalities (QVI) in this context, with the standard formulation of dynamic programming. We show that the QVI formulation brings many interesting features, completing the standard formulation of dynamic programming. We study applications to inventory control in discrete time.

We consider mixed boundary value problems for a piezoelectric medium with a crack. The study is based on Mindlin's model of piezoelectricity which takes into account the influence of the polarization gradient. Using the potential methods and the theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions and establish their regularity properties.

In this paper we use some energy methods to study the location (and formation) of a free boundary arising in some unilateral problems, for instance, in the obstacle problem and the Stefan problem .

We present a different method for studying the Weingarten map for a hypersurface in the Euclidean space . Applying the Cartesian coordinates of the ambient space and tangential Günter's derivatives we obtain a simple matrix representation formula for the Weingarten map for implicit hypersurfaces, which can be applied, for example, to calculate the mean and Gauß's curvatures without passing to intrinsic coordinates.

We consider the application of the concept of nonlinear capacity induced by nonlinear operators to blow-up problems for various types of nonlinear partial differential equations involving equations with nonlocal nonlinearities.

We prove the existence of a strong corrector for the linearized incompressible Navier–Stokes solution on a domain with characteristic boundary. This case is different from the noncharacteristic case considered in [Hamouda and Temam, Some singular perturbation problems related to the Navier–Stokes equations: Springer Verlag, 2006] and somehow physically more relevant. More precisely, we show that the linearized Navier–Stokes solutions behave like the Euler solutions except in a thin region, close to the boundary, where a certain heat equation solution is added (the corrector). Here, the Navier–Stokes equations are considered in an infinite channel of but our results still hold for more general bounded domains.

Large time behavior of solutions to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. The rate of convergence is given, too. Dirichlet boundary conditions with homogeneous data are considered.

One boundary value problem for a hyperbolic equation with power nonlinearity and the iterated wave operator in the principal part is considered in a conical domain. Depending on the index of nonlinearity and spatial dimensionality of the equation the question on the existence and uniqueness of a solution of a boundary value problem is investigated. The question as to the absence of a solution of this problem is also considered.

In the rectangle Ω = [0, a ] × [0, b ] the nonlinear hyperbolic equation 𝑢 (2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

The known theory on the one-side exact boundary observability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see [Tatsien, C. R. Math. Acad. Sci. 342: 937–942, 2006]–[Tatsien, ESAIM Control Optim. Calc. Var.], [Russell, SIAM Rev. 20: 639–739, 1978]). In this paper we illustrate, with an inspiring example, that the one-side exact boundary observability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary conditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

We derive some formulas for the critical crossing, also called the principle of exchange of stability (PES) in physics. These formulas will be crucial for studying dynamic transitions of many problems in science and engineering [Ma and Wang, Bifurcation theory and applications, World Scientific Publishing Co. Pte. Ltd., 2005, Ma and Wang, Stability and bifurcation of nonlinear evolutions equations, Science Press, 2007].