scholarly journals GLOBAL WELL-POSEDNESS OF THE PERIODIC CUBIC FOURTH ORDER NLS IN NEGATIVE SOBOLEV SPACES

2018 ◽  
Vol 6 ◽  
Author(s):  
TADAHIRO OH ◽  
YUZHAO WANG
Keyword(s):  
Fourth Order ◽  
Sobolev Spaces ◽  
Energy Functional ◽  
Well Posedness ◽  
Initial Condition ◽  
Energy Functionals ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
The Difference ◽  
Negative Sobolev Spaces

We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces $H^{s}(\mathbb{T})$, $s>-\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in $H^{s}(\mathbb{T})$, $s>-\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo–Oh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in $H^{s}(\mathbb{T})$, $s>-\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the $H^{s}$-energy functional, allowing us to introduce an infinite sequence of correction terms to the $H^{s}$-energy functional in the spirit of the $I$-method. In fact, the main novelty of this paper is this reduction of the $H^{s}$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

scholarly journals Higher-order Lagrangian perturbative theory for the Cosmic Web

2014 ◽  
Vol 11 (S308) ◽  
pp. 119-120
Author(s):  
Takayuki Tatekawa ◽  
Shuntaro Mizuno
Keyword(s):  
Perturbation Theory ◽  
Fourth Order ◽  
Initial Conditions ◽  
Higher Order ◽  
Order Perturbation ◽  
Initial Condition ◽  
The Difference ◽  
Fifth Order ◽  
The Universe ◽  
Higher Order Lagrangian

AbstractZel'dovich proposed Lagrangian perturbation theory (LPT) for structure formation in the Universe. After this, higher-order perturbative equations have been derived. Recently fourth-order LPT (4LPT) have been derived by two group. We have shown fifth-order LPT (5LPT) In this conference, we notice fourth- and more higher-order perturbative equations. In fourth-order perturbation, because of the difference in handling of spatial derivative, there are two groups of equations. Then we consider the initial conditions for cosmological N-body simulations. Crocce, Pueblas, and Scoccimarro (2007) noticed that second-order perturbation theory (2LPT) is required for accuracy of several percents. We verify the effect of 3LPT initial condition for the simulations. Finally we discuss the way of further improving approach and future applications of LPTs.

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

2018 ◽  
Vol 15 (03) ◽  
pp. 463-557 ◽  
Author(s):  
Chulkwang Kwak
Keyword(s):  
Kdv Equation ◽  
Main Tool ◽  
Well Posedness ◽  
Current Form ◽  
Low Regularity ◽  
Integrable Structure ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
Modified Kdv Equation ◽  
Fifth Order

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in [Formula: see text], [Formula: see text], via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173(2) (2008) 265–304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract 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.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

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