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We investigate the evolution problem where H is a Hilbert space, A is a self-adjoint nonnegative operator on H with domain D (A), delta > 0 is a parameter, and m(r) is a nonnegative function such that m(0) = 0 and m is nonnecessar...
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We investigate the evolution problem where H is a Hilbert space, A is a self-adjoint nonnegative operator on H with domain D (A), delta > 0 is a parameter, and m(r) is a nonnegative function such that m(0) = 0 and m is nonnecessarily Lipschitz continuous in a neighborhood of 0. We prove that this problem has a unique global solution for positive times., provided that the initial data (u(0), u(1)) is an element of D(A) x D(A(1/2)) satisfy a suitable smallness assumption and the nondegeneracy condition m(vertical bar A(1/2)u(0)vertical bar(2)(H)) > 0. Moreover, we study the decay of the solution as t -> +infinity. These results apply to degenerate hyperbolic PDEs with nonlocal nonlinearities.
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Let H be a real Hilbert space, with norm |·) and scalar product <·,·>. Let A be a self-adjoint linear coercive operator on H with dense domain D(A).
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We combine the methods of the topological degree with techniques developed by J. K. Hale and G. Raugel to study the periodic solutions of a damped nonlinear hyperbolic equation in a thin domain. (C) 1997 Academic Press.
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We investigate the evolution problem u″ + δu′ + m(|A~(1/2)u|~2)Au = 0, u(0) = u_0, u′ (0) = u_1, where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), δ > 0 is a parameter, and m: [0, +...
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We investigate the evolution problem u″ + δu′ + m(|A~(1/2)u|~2)Au = 0, u(0) = u_0, u′ (0) = u_1, where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), δ > 0 is a parameter, and m: [0, +∞[ →[0, +∞[ is a locally Lipschitz continuous function. We prove that this problem has a unique global solution for positive times, provided that the initial data (u_0, u_1) ∈ D(A) x D(A~(1/2)) satisfy a suitable smallness assumption and the non-degeneracy condition m(|A~(1/2)u_0|~2) > 0. Moreover (u(t), u′(t),u″(t))→(u_∞,0,0) in D(A) x D(A~(1/2)) x H as t → +∞, where |A~(1/2)u_∞|m(|A~(1/2)u_∞|~2) = 0. These results apply to degenerate hyperbolic PDEs with non-local non-linearities.
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The moment method is a successful way to approximate the solution of the Boltzmann equation in reasonable runtime with relatively few unknowns while allowing for non-equilibrium effects of the gas. However, the convergence of the ...
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The moment method is a successful way to approximate the solution of the Boltzmann equation in reasonable runtime with relatively few unknowns while allowing for non-equilibrium effects of the gas. However, the convergence of the moment method with respect to an increasing number of moments is typically slow. This paper aims to improve the convergence of the moment method by introducing filtered hyperbolic moment equations that result in virtually no additional computational overhead while significantly reducing the error. The filter approach is based on a careful study of averaging solutions of two adjacent moment systems and the reformulation of the averaging using an artificial collision method that naturally gives rise to the filter. We study the properties of the filter and show numerical test cases of one-dimensional problems that demonstrate the superior quality of the new filtered moment method.
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We consider. degenerate Kirchhoff equations with a small parameter. ε in front of the second-order time-derivative. It is well known that these equations admit global solutions when. ε is small enough, and that these solutions d...
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We consider. degenerate Kirchhoff equations with a small parameter. ε in front of the second-order time-derivative. It is well known that these equations admit global solutions when. ε is small enough, and that these solutions decay as. t→. +. ∞ with the same rate of solutions of the limit problem (of parabolic type).In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as ε→0 ~+, and as t→+∞. Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as t→+∞).Proofs involve a nonlinear step where we separate Fourier components with respect to the lowest frequency, followed by a linear step where we exploit weighted versions of classical energies.
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We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter ε in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates...
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We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter ε in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as ε→0 ~+, and as t→+∞. In particular, in that case the difference decays faster than the two terms separately. In this paper we consider the complementary assumption on initial data, and we show that now the optimal decay-error estimates involve a decay rate which is slower than the decay rate of the two terms. In both cases, the improvement or deterioration of decay rates depends on the smallest frequency represented in the Fourier components of initial data.
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Degeneration in a small spatial coordinate of the original initial-boundary-value problem for hyperbolic equations is considered in order to construct simpler models. The problem is investigated in a domain whose one spatial scale...
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Degeneration in a small spatial coordinate of the original initial-boundary-value problem for hyperbolic equations is considered in order to construct simpler models. The problem is investigated in a domain whose one spatial scale is much less than the other scales. This makes it possible to expand the desired functions in power series and hence to reduce the dimension of the problem. However, this comes at the cost of degeneration of the spectrum of the original three-dimensional problem.
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We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibilit...
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We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.
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In this paper we study the Cauchy problem for hyperbolic equations in the setting of Hormander S(m, g) classes. We provide regularity estimates, existence and uniqueness in the scale of Sobolev spaces H (m, g) adapted to the Weyl-...
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In this paper we study the Cauchy problem for hyperbolic equations in the setting of Hormander S(m, g) classes. We provide regularity estimates, existence and uniqueness in the scale of Sobolev spaces H (m, g) adapted to the Weyl-Hormander calculus. We also obtain estimates for some parabolic evolution equations. (C) 2015 Elsevier Inc. All rights reserved.
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