Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media

Yang, Shiwu

doi:10.1007/s00205-013-0631-y

Cited by 48 publications

(103 citation statements)

References 30 publications

(62 reference statements)

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“…However, as in [30], the same conclusion holds on curved background (R 3+1 , g) with metric g merely C 1 close to the Minkowski metric and coinciding with the Minkowski metric outside the cylinder {(t, x)||x| ≤ R}.…”

Section: Remarkmentioning

confidence: 71%

“…Our argument for the main Theorem 2 is similar to that in [30], which relies on a new approach, originally developed by Dafermos-Rodnianski [6]. This new approach is a combination of an integrated local energy inequality obtained by using the vector field f (r )∂ r as multipliers and a p-weighted energy inequality derived by using r p (∂ t + ∂ r ) as multipliers localized in a neighborhood of the null infinity.…”

Section: Remarkmentioning

confidence: 99%

“…This new approach is a combination of an integrated local energy inequality obtained by using the vector field f (r )∂ r as multipliers and a p-weighted energy inequality derived by using r p (∂ t + ∂ r ) as multipliers localized in a neighborhood of the null infinity. However, due to the weak decay of ∂ , we are not able to obtain the integrated local energy inequality and the p-weighted energy inequalities separately as in [30]. We thus consider these two inequalities together, see Proposition 1 in Sect.…”

Section: Remarkmentioning

confidence: 99%

“…These two estimates, which the decay of the energy flux E[φ](τ ) relies on, were shown separately in [30]. Due to the weak decay of the functions (t, x), L μ (t, x), we are not able to show these two estimates separately.…”

Section: Weighted Energy Inequalitiesmentioning

confidence: 99%

“…In this paper, we use the approach developed in [6,30] to treat the problem of global stability of solutions to nonlinear wave equations. This new method avoids the use of vector fields containing positive weights in t, e.g., the scaling vector field S = t∂ t +r ∂ r and only requires commutation with the angular momentum and the generator of the time translation.…”

Section: Introductionmentioning

confidence: 99%

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Global stability of solutions to nonlinear wave equations

Yang

2014

Sel. Math. New Ser.

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We consider the problem of global stability of solutions to a class of semilinear wave equations with null condition in Minkowski space. We give sufficient conditions on the given solution, which guarantees stability. Our stability result can be reduced to a small data global existence result for a class of semilinear wave equations with linear terms B μν ∂ μ (t, x)∂ ν φ, L μ (t, x)∂ μ φ and quadratic terms h μν (t, x)∂ μ φ∂ ν φ where the functions (t, x), L μ (t, x), h μν (t, x) decay rather weakly and the constants B μν satisfy the null condition. We show the small data global existence result by using the new approach developed by Dafermos-Rodnianski. In particular, we prove the global stability result under weaker assumptions than those imposed by Alinhac (Indiana Univ Math J 58 (6): 2009).

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Section: Remarkmentioning

confidence: 71%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Weighted Energy Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global stability of solutions to nonlinear wave equations

Yang

2014

Sel. Math. New Ser.

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Some remarks on quasilinear wave equations with null condition in 3‐D

Zha

2016

Math Methods in App Sciences

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In this paper, we give an alternative proof to the global existence result, which is originally owing to the pioneering work of Klainerman and Christodoulou, for the Cauchy problem of quasilinear wave equations with null condition in three space dimensions. The proof can display the following three features simultaneously: the Lorentz boost operator is not employed in the generalized energy estimates; the generalized energy of the solution will be always small, which was first observed by Alinhac; and the initial data are not assumed to have a compact support. Copyright © 2016 John Wiley & Sons, Ltd.Keywords: quasilinear wave equations; three space dimensions; null condition; global existence IntroductionWe consider the following quasilinear wave equation in 3-D:where D @ 2 t is the wave operator,3 /, and the coefficients g˛ˇare smooth real functions vanishing at the origin, that is,We always assume that the following symmetric conditions hold:which is an essential condition for energy estimate in our proof. The set g D .g˛ˇ / is said to satisfy the null condition if for any givenConsider the Cauchy problem for (1) with the initial dataFor the Cauchy problems (1) and (5), under the null condition (4), the global existence of classical solutions with small initial data was first proved by Christodoulou [1] and Klainerman [2] independently. Christodoulou's proof employed the conformal transformation method, while Klainerman used the vector fields method. Klainerman's proof relies on a pair of coupled differential inequalities for the lower-order weighted L 1 norm of the solution and the higher-order energy. It is worth noting that the higher-order energy may be not always small and may grow with the time.In [3] (Section 9.1.3), Alinhac got the global existence of classical solutions to the Cauchy problems (1) and (5) via a single generalized energy estimate, and the generalized energy can be shown to be always small. The key point of his proof is the ghost weight energy We note that in both Alinhac's proof and Klainerman's proof, they all use the full Lorentz invariance of the wave operator; therefore, the general space-time derivatives, spatial rotation, scaling and Lorentz boost operator are all used. Moreover, in Alinhac [3], it is essentially assumed that the initial data have a compact support. However, the Lorentz boost operator is not suitable for some wave systems, which are not Lorentz invariant, such as nonrealistic system with multiple wave speeds [7][8][9], nonlinear wave equations on non-flat space-time [10-13], wave-type equation with nonlocal term [14,15] and exterior problem [16]. Furthermore, the compact support assumption of the initial data is also restrictive and is not suitable for some nonlocal problems such as the incompressible elastodynamics(e.g. Sideris and Thomases [14,15], Lei and Wang [17], Lei [18] and Wang [19]). So how to remove the Lorentz boost operator and the compact support assumption of the initial data in Alinhac's proof, and in the meantime, keep the smallness of the general...

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