The classical wave equation in the space of generalized functions (distributions)is considered. The Distributions Method of building the solutions of nonstationary boundary value problems (NBVP) for wave equations in coordinates spaces of different dimensions is elaborated. Dynamic analogues of Green and Gauss formulas for solutions of wave equation in the distributions space are built. Their regular integral representations and singular boundary integral equations for solving the nonstationary problems are constructed for the coordinates spaces of the 1,2,3-dimensions. The method of obtaining of conditions on shock waves fronts is stated. The theorem on the single solution of NBVP is proved, including shock waves.
We consider two dimensional nonstationary scattering of plane waves by a NN-wedge. We prove the existence and uniqueness of a solution to the corresponding mixed problem and we give an explicit formula for the solution. Also the Limiting Amplitude Principle is proved and a rate of convergence to the limiting amplitude is obtained.
We study linear and quasilinear Venttsel initial-boundary value problems for parabolic operators with discontinuous coefficients. On the basis of the a priori estimates obtained, strong solvability in composite Sobolev spaces is proved.
In this paper we prove the existence and uniqueness of a solution to the nonstationary two dimensional system of equations describing miscible liquids with nonsmooth, multivalued and nonmonotone boundary conditions of subdifferential type. We employ the regularized Galerkin method combined with results from the theory of hemivariational inequalities.
We deal with the Cauchy problem for a perturbed wave equation in the half-plane with data given on a part of the space-time boundary. The equation in consideration describes a wave process in a laterally inhomogeneous medium. We propose a reconstruction algorithm, which is applicable to the problem of determining nonstationary wave field from boundary data arising in geophysics.
We consider the Cauchy problem for the wave equation in $Ω\times{\mathbb R}$ with data given on some part of the boundary $\partialΩ\times{\mathbb R}$. We provide a reconstruction algorithm for this problem based on analytic expressions. Our result is applicable to the problem of determining nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.
We consider two-dimensional nonstationary Navier-Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.
This second article in a two-part series (following [arXiv:2105.02865], listed here as \cite{L}) proves optimal pointwise decay rates for the quintic defocusing wave equation with large initial data on nonstationary spacetimes, and both the quintic defocusing and quintic focusing wave equations with small initial data on nonstationary spacetimes. We prove a weighted local energy decay estimate, and use local energy decay and Strichartz estimates on these variable-coefficient backgrounds. By using an iteration scheme, we obtain the optimal pointwise bounds. In addition, we explain how the itera…
In the paper we prove the existence results for initial-value boundary value problems for compressible isothermal Navier-Stokes equations. We restrict ourselves to 2D case of a problem with no-slip condition for nonstationary motion of viscous compressible isothermal fluid. However, the technique of modeling and analysis presented here is general and can be used for 3D problems.
In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The direct problem is an initial-boundary value problem for a system for the velocity field, the angular velocity of rotation of the fluid particles, the mass density and the pressure distribution. The inverse problem consists in the external force recover assuming an integral measurements on the boundary. We characterize the inverse problem solutions using an operator equation of second kind, which is…
We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we…
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } b \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $. In particular, compared with the…
We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the $\mathscr{R}$-boundedness of the solution operator families with resolvent parameter $λ$ varying in a sector $Σ_{\varepsilon,γ_0}$ for any $γ_0>0$ and $0<\varepsilon<π/2$, whe…
We study the long time existence of solutions to nonlinear wave equations with power-type nonlinearity (of order $p$) and small data, on a large class of $(1+n)$-dimensional nonstationary asymptotically flat backgrounds, which include the Schwarzschild and Kerr black hole space-times. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we give lower bounds of the lifespan when $n=3, 4$ and $p$ is not bigger than the critical one. The lower bounds for three dimensional subcritical and four dimensional critical cases are sharp in genera…
In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.
In this paper, we consider the stabilization of wave equations with moving boundary. First, we show the solution behaviour of wave equation with Neumann boundary conditions, that is, the energy of wave equation with mixed boundary conditions may decrease, increase or conserve depending on the different range of parameter. Second, we prove the wellposedness and stabilization for the wave equation with time delay and moving boundary.
