Equation In Besov Spaces Research Articles

Non-uniform dependence on initial data for the Euler equations in Besov spaces

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21].

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Global well-posedness and large time behavior for the 3D magneto-micropolar equations

This paper focuses on the global well-posedness and optimal decay estimates to the 3D magneto-micropolar equations in Besov spaces. We obtain the global existence and uniqueness of solutions in Besov spaces. Moreover, the decay estimates of the global solutions are established provided that the initial data belongs to the negative Besov space.

Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces

Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces

Zero-filter limit issue for the Camassa–Holm equation in Besov spaces

Zero-filter limit issue for the Camassa–Holm equation in Besov spaces

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

Analysis on continuity of the solution map for the Whitham equation in Besov spaces

AbstractThis paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces () on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23].

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces

Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces

Ill-posedness for the Euler–Poincaré equations in Besov spaces

We prove that the initial value problem for the Euler–Poincaré equations is not locally well-posed for initial data in the Besov space Bp,∞σ whenever σ>2+max{1+dp,32}. By presenting a new construction of initial data u0, we prove the corresponding solution of Euler–Poincaré equations starting from u0 is discontinuous at t=0 in the norm of Bp,∞σ, which implies the ill-posedness. Since this problem is locally well-posed in the Besov space Bp,rσ for r<∞ and the same σ, our result suggests that well-posedness does not hold at the endpoint r=∞.

Ill-posedness for the Euler equations in Besov spaces

In the paper, we consider the Cauchy problem to the Euler equations in Rd with d≥2. We construct an initial data u0∈Bp,∞σ showing that the corresponding solution map of the Euler equations starting from u0 is discontinuous at t=0 in the metric of Bp,∞σ, which implies the ill-posedness for this equation in Bp,∞σ. We generalize the periodic result of Cheskidov and Shvydkoy (2010).

On the Continuity of the Solution Map of the Euler–Poincaré Equations in Besov Spaces

By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in $$B^s_{p,r}(\mathbb {R}^d)$$ with $$s>\max \{1+\frac{d}{2},\frac{3}{2}\}$$ and $$(p,r)\in (1,\infty )\times [1,\infty )$$ . This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of $$B^s_{p,r}(\mathbb {R}^d)$$ near the origin.

A PRIORI ESTIMATES FOR THE FIFTH-ORDER MODIFIED KDV EQUATIONS IN BESOV SPACES WITH LOW REGULARITY

We get a priori estimates for the fifth-order modified KdV equations in Besov spaces with low regularity which cover the full subcritical range. These estimates are obtained from the power series expansion of the perturbation determinant associated to the Lax pair. More precisely, we get the global in time bounds of the $B^s_{2,r}$ norm of the solution for $-1/2&lt; s &lt; 1$, $1\leq r\leq \infty$. Then we can obtain the sharp global well-posedness in $H^s$ for $s\geq 3/4$, which is the minimal regularity threshold for which the well-posedness problem can be solved via the contraction principle.

A Priori Estimates for the Derivative Nonlinear Schrödinger Equation

We prove low regularity a priori estimates for the derivative nonlinear Schr\"{o}dinger equation in Besov spaces with positive regularity index conditional upon small $L^2$-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip--Vi\c{s}an--Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.

The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg–Whitham equation in Besov spaces

In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg–Whitham equation in both supercritical Besov spaces Bp,rs,s>1+1p,1≤p,r≤+∞ and critical Besov spaces Bp,11+1p,1≤p<+∞, which improves the previous work (Holmes and Thompson, 2017; Holmes, 2016; Yin, 2007). Then, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces Bp,rs,s>1+1p,1≤p≤+∞,1≤r<+∞ or s=1+1p,1≤p<+∞,r=1. At last, we show that the Cauchy problem for the Fornberg–Whitham equation is ill-posed in Bp,∞σ with σ>3+1p,1≤p≤+∞.

Non-uniform Continuity of the Generalized Camassa–Holm Equation in Besov Spaces

In the paper, we gave a strengthening of our previous work in \cite{Li1} (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.

On the Cauchy problem for a weakly dissipative Camassa-Holm equation in critical Besov spaces

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces with and Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.

Global existence for the Jordan–Moore–Gibson–Thompson equation in Besov spaces

In this paper, we consider the Cauchy problem of a model in nonlinear acoustic, named the Jordan–Moore–Gibson–Thompson equation. This equation arises as an alternative model to the well-known Kuznetsov equation in acoustics. We prove global existence and optimal time decay of solutions in Besov spaces with a minimal regularity assumption on the initial data, lowering the regularity assumption required in Racke and Said-Houari (Commun Contemp Math. 1–39, 2019. https://doi.org/10.1142/S0219199720500698) for the proof of the global existence. Using a time-weighted energy method with the help of appropriate Lyapunov-type estimates, we also extend the decay rate in Racke and Said-Houari (2019) and show an optimal decay rate of the solution for initial data in the Besov space \(\dot{{B}}_{2,\infty }^{-3/2}(\mathbb {R}^3)\), which is larger than the Lebesgue space \(L^1(\mathbb {R}^3)\) due to the embedding \(L^1(\mathbb {R} ^3)\hookrightarrow \dot{{B}}_{2,\infty }^{-3/2}(\mathbb {R}^3)\). Hence, we removed the \(L^1\)-assumption on the initial data required in Racke and Said-Houari (2019) in order to prove the decay estimates of the solution.

Sharp ill-posedness for the generalized Camassa–Holm equation in Besov spaces

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation that containing, as its members, three integrable equations: the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa–Holm equation in $$B^s_{p,\infty }$$ with $$s>\max \{1+1/p, 3/2\}$$ and $$1\le p\le \infty $$ in the sense that the solution map to this equation starting from $$u_0$$ is discontinuous at $$t = 0$$ in the metric of $$B^s_{p,\infty }$$ . Our result covers and improves the previous work given in Li et al. (J Differ Equ 306:403–417, 2022), solving an open problem left in Li et al. (2022).

The Cauchy Problem and Multi-peakons for the mCH-Novikov-CH Equation with Quadratic and Cubic Nonlinearities

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.