We prove the existence of global weak solution of the two-dimensional dissipative quasi-geostrophic equations with small initial data in M-p,sigma(s) and local well-posedness with the large initial data in the same sp...
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We prove the existence of global weak solution of the two-dimensional dissipative quasi-geostrophic equations with small initial data in M-p,sigma(s) and local well-posedness with the large initial data in the same space. Our proof is based on constructing a commutator related to the problem, as well as its estimate. Copyright (C) 2017 John Wiley & Sons, Ltd.
We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especial...
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We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especially, we establish the analogue of the sharp Sobolev embedding theorem on weighted modulation spaces.
We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed. the Weyl product is equivalent to a matrix multiplication on the Gabor ...
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We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed. the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sjostrand space M omega(infinity,1), then the matrix has polynomial or exponential off-diagonal decay, depending oil the weight omega. Moreover, if its operator is invertible on L-2, the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time-frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter. (C) 2008 Elsevier Inc. All tights reserved.
We consider the nonlinear Schrodinger equation { iu(t) + Delta u +/- f(u) = 0, f(u) = vertical bar u vertical bar(2m) u(n,) m, n is an element of N, (x, t) is an element of R-N X R u(0, x) = u(0)(x). (*) We give a rep...
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We consider the nonlinear Schrodinger equation { iu(t) + Delta u +/- f(u) = 0, f(u) = vertical bar u vertical bar(2m) u(n,) m, n is an element of N, (x, t) is an element of R-N X R u(0, x) = u(0)(x). (*) We give a representation of the unique solution of the Cauchy problem (*) existed in C(0, T*;M-p,M-1). Moreover, by this representation, we obtain that there exists a constant B independent of p, N such that for any initial data parallel to u(0)parallel to M-p',M-1< B(N(1/2 - 1/p) - 1), A 2N/N-2 < p <= 2m + n + 1, N > 2 p <= 2m + n + 1, N > 2, the Schrodinger equation (*) has a unique global solution u is an element of C(-infinity, infinity;M-p,M-1). (C) 2015 Elsevier Ltd. All rights reserved.
In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local...
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In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local solutions and global solutions. Then scattering and stability of solutions are also discussed. The proof is mainly based on the decay properties of the solutions operator in modulation spaces and the contraction mapping principle.
Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a ...
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Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Benyi et al., see [A. Benyi, L. Grafakos, K. Grochenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131-139]. Finally, we show that any function with ([d/2] + 1)-times bounded derivatives is a Fourier multiplier for all modulation spaces M-p,M-q (R-d) with p is an element of (1, infinity) and q is an element of [1, infinity]. (c) 2006 Elsevier Inc. All rights reserved.
We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on L-p-spaces, ...
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We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on L-p-spaces, Sobolev spaces, and modulation spaces.
We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u) = (K * vertical bar u vertical bar(2))u under a specified condition on potential K with Cauchy data in modulation spaces M-p,M-...
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We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u) = (K * vertical bar u vertical bar(2))u under a specified condition on potential K with Cauchy data in modulation spaces M-p,M-q(R-d). We establish global well-posedness results in M-1,M-1(R-d) when K(x) = lambda vertical bar x vertical bar(-gamma)(lambda subset of R, 0 < gamma < min{2, d/2});in M-p,M-d(R-d) (1 <= q <= min{p, p'} where p' is the Holder conjugate of p is an element of [1, 2]) when K is in Fourier algebra FL1 (R-d), and local well-posedness result in M-p,M-1 (R-d) (1 <= p <= infinity) when K is an element of M-1,M-infinity (R-d). (C) 2015 Elsevier Ltd. All rights reserved.
modulation spaces M-p,q(s) were introduced by Feichtinger [11] in 1983. Benyi and Oh [2] defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possibl...
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modulation spaces M-p,q(s) were introduced by Feichtinger [11] in 1983. Benyi and Oh [2] defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with Benyi and Oh's modulation spaces, we introduce the scaling limit versions of modulation spaces, which contains both Feichtinger's and Benyi and Oh's modulation spaces. As their applications, we will give a local well-posedness and a (small data) global well-posedness results for nonlinear Schrodinger equation in some scaling limit of modulation spaces, which generalize the well posedness results of [3] and [23], and certain super-critical initial data in H-s or in L-p are involved in these spaces. (C) 2021 Elsevier Inc. All rights reserved.
We show that the Gelfand - Shilov algebra S S-1/2(1/2) is densely embedded in the weighted modulation space M-m(1). Here the weight function m is allowed to have a super- exponential growth at infinity. The basic tool...
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We show that the Gelfand - Shilov algebra S S-1/2(1/2) is densely embedded in the weighted modulation space M-m(1). Here the weight function m is allowed to have a super- exponential growth at infinity. The basic tool is given by an integral transform called short- time Fourier transform (STFT). The STFT is used to both define and characterize the previous spaces. Moreover, our result is attained using the properties of the STFT and its adjoint.
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