Periodic Weak Solutions Research Articles

Boundary value problem for the three dimensional time periodic Vlasov-Maxwell system

In this work we study the existence of time periodic weak solution for the three dimensional Vlasov-Maxwell system with boundary conditions. The main idea consists of using the mass, momentum and energy conservation laws which allow us to obtain a priori estimates in the case of a star-shaped bounded spatial domain. We start by constructing time periodic smooth solutions for a regularized system. The existence for the Vlasov-Maxwell system follows by weak stability under uniform estimates. These results apply for both classical and relativistic cases and for systems with several species of particles.

Positive doubly periodic solutions of nonlinear telegraph equations

This paper deals with the existence of nontrivial doubly periodic solutions for the nonlinear telegraph equation u tt−u xx+cu t+a(t,x)u=b(t,x)f(t,x,u), (t,x)∈ R 2, where c>0 is a constant, a,b∈C( R 2, R +) , f∈C( R 2× R +, R +) , and a, b, f are 2 π-periodic in t and x. We show the existence of positive doubly periodic weak solutions when 0⩽ a( t, x)⩽ c 2/4 and f is either superlinear or sublinear on u by using the Krasnoselskii's fixed point theorem in cones.

Maxwell's equations in periodic chiral structures

AbstractConsider a time–harmonic electromagnetic plane wave incident on a biperiodic structure in ℝ3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and heterogeneous. In general, wave propagation in the chiral medium is governed by Maxwell's equations together with the Drude– Born–Fedorov (constitutive) equations. In this paper, the diffraction problem is formulated in a bounded domain by introducing a pair of transparent boundary conditions. It is then shown that for all but possibly a discrete set of parameters, there is a unique quasi–periodic weak solution to the diffraction problem. Our proof is based on the Hodge decomposition, a compact imbedding result, and the Lax–Milgram Lemma. In addition, an energy conservation for the weak solution is also shown.

Reproductive weak solutions of magneto-micropolar fluid equations in exterior domains

We establish the existence of a reproductive weak solution, a so-called periodic weak solution, for the equations of motion of magneto-micropolar fluids in exterior domains in R 3.

Perturbed S 1-symmetric hamiltonian systems

By techniques of critical point theory, we show that multiple periodic weak solutions of a general class of Hamiltonian systems persist despite perturbation with an L 2 term destroying the S 1-symmetries.

PERIODIC SOLUTIONS OF THE VLASOV–POISSON SYSTEM WITH BOUNDARY CONDITIONS

We study the Vlasov–Poisson system with time periodic boundary conditions. For small data we prove existence of weak periodic solutions in any space dimension. In the one-dimensional case the result is stronger: we obtain existence of mild solution and uniqueness of this solution when the data are smooth. It is necessary to impose a nonvanishing condition for the incoming velocities in order to control the lifetime of particles in the domain.

Existence and uniqueness of periodic solutions for a nonlinear reaction-diffusion problem

We consider a class of degenerate reaction-diffusion equations on a bounded domain with nonlinear flux on the boundary. These problems arise in the mathematical modelling of flow through porous media. We prove, under appropriate hypothesis, the existence and uniqueness of the nonnegative weak periodic solution. To establish our result, we use the Schauder fixed point theorem and some regularizing arguments.

Periodic solutions of the 1D Vlasov-Maxwell system with boundary conditions

We study the ID Vlasov-Maxwell system with time-periodic boundary conditions in its classical and relativistic form. We are mainly concerned with existence of periodic weak solutions. We shall begin with the definitions of weak and mild solutions in the periodic case. The main mathematical difficulty in dealing with the Vlasov-Maxwell system consist of establishing L∞ estimates for the charge and current densities. In order to obtain this kind of estimates, we impose non-vanishing conditions for the incoming velocities, which assure a finite lifetime of all particles in the computational domain ]0, L[. The definition of the mild solution requires Lipschitz regularity for the electro-magnetic field. It would be enough to have a generalized flow but the result of DiPerna Lions (Invent. Math. 1989; 98: 511-547) does not hold for our problems because of boundary conditions. Thus, in the first time, the Vlasov equation has to be regularized. This procedure leads to the study of a sequence of approximate solutions. In the same time, an absorption term is introduced in the Vlasov equation, which guarantees the uniqueness of the mild solution of the regularized problem. In order to preserve the periodicity of the solution, a time-averaging vanishing condition of the incoming current is imposed: ∫ T 0 dt ∫ r ∫C x go(t, r x , v x )dv+∫ t 0 dt∫ rx ∫v x g z (t,c x ,v y )dv = 0 (1) where g O . g L are the incoming distributions f(t, 0, v x , v y ) = g O (t, v x , v y ), t ∈ R t , v x > 0, v y ∈ R v (2) f(t, L, v x , v y ) = g L (t, v x , v y ), t ∈ R t , v x < 0, v y ∈ R v (3) The existence proof uses the Schauder fixed point theorem and also the velocity averaging lemma of DiPerna and Lions (Comm. Pure Appl. Math. 1989; XVII: 729-757). In the last section we treat the relativistic case.

Weak periodic solutions of non-Newtonian viscous incompressible fluids

The existence of weak periodic solution in time of non-Newtonian fluids with a time periodic forcing is established.

Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains

For an arbitrary domain Ω ⊂ ℝn, n=2,3, Ω ≠ ℝn, we prove the existence of weak periodic solutions to the Navier-Stokes equations and of regular solutions if the data are small or satisfy certain symmetry conditions. We also show that the periodic regular solutions are stable. Bibliography: 38 titles.

Discontinuous wave equations and a topological degree for some classes of multi-valued mappings

The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.

Existence and regularity of weak periodic solutions of the 2-D wave equation

Existence and regularity of weak periodic solutions of the 2-D wave equation

Existence of periodic weak solutions to viscoelastic dynamic equation with memory

Existence of periodic weak solutions to viscoelastic dynamic equation with memory

The point-vortex method for periodic weak solutions of the 2-D Euler equations

Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or Lp with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H−1. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed.

Existence of Almost Periodic Weak Type Solutions for the Conservative Forced Pendulum Equation

Existence of Almost Periodic Weak Type Solutions for the Conservative Forced Pendulum Equation

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of\(W_p^{2\beta }\)-periodic weak solutions under some reasonable assumptions.

On the existence of periodic weak solutions of Navier-Stokes equations in regions with periodically moving boundaries

We prove the existence of periodic weak solutions to the Navier-Stokes equations in regions with moving boundaries using the elliptic regularization.

Almost periodic weak solutions to forced pendulum type equations without friction

We obtain sufficient conditions for the existence of almost periodic weak solutions to equations of the frictionless pendulum type with forcing.

Almost periodic weak solutions to forced pendulum type equations without friction

Almost periodic weak solutions to forced pendulum type equations without friction

Time—periodic weak solutions

In continuing from previous papers, where we studied the existence and uniqueness of the global solution and its asymptotic behavior as time t goes to infinity, we now search for a time‐periodic weak solution u(t) for the equation whose weak formulation in a Hilbert space H is urn:x-wiley:01611712:media:ijmm464936:ijmm464936-math-0001 where: ′ = d/dt; (′) is the inner product in H; b(u, v), a(u, v) are given forms on subspaces U ⊂ W, respectively, of H; δ &gt; 0, α ≥ 0, β ≥ 0 are constants and α + β &gt; 0; G is the Gateaux derivative of a convex functional J : V ⊂ H → [0, ∞) for V = U, when α &gt; 0 and V = W when α = 0, hence β &gt; 0; v is a test function in V; h is a given function of t with values in H.Application is given to nonlinear initial‐boundary value problems in a bounded domain of Rn.