Sharp Well-posedness Research Articles

Sharp well-posedness for the Benjamin–Ono equation

Sharp well-posedness for the Benjamin–Ono equation

Sharp well-posedness for the cubic NLS and mKdV in

Abstract We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$ . Well-posedness has long been known for $s\geq 0$ , see [55], but not previously for any $s<0$ . The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in $H^s({{\mathbb {R}}})$ for any $s>-\frac 12$ . The best regularity achieved previously was $s\geq \tfrac 14$ (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.

Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane

Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane

Sharp well-posedness and blowup results for parabolic systems of the Keller–Segel type

Sharp well-posedness and blowup results for parabolic systems of the Keller–Segel type

Rough solutions of the 3-D compressible Euler equations

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho\mathfrak{w} \in H^s \times H^s \times H^{s'}$, $2\lt s'\lt s$. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., $\mathfrak{w}=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for $\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$. We therefore, conjecture that the optimal result should be $(v,\varrho,\mathfrak{w} \in H^s\times H^s\times H^{s'}$, $s>2$, $s' > \frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustical spacetime.

KdV on an incoming tide

Given smooth step-like initial data V(0, x) on the real line, we show that the Korteweg–de Vries equation is globally well-posed for initial data . The proof uses our general well-posedness result (2021 arXiv:2104.11346). As a prerequisite, we show that KdV is globally well-posed for perturbations of step-like initial data. In the case V ≡ 0, we obtain a new proof of the Bona–Smith theorem (Bona and Smith 1975 Trans. R. Soc. A 278 555–601) using the low-regularity methods that established the sharp well-posedness of KdV in H −1 (Killip and Vişan 2019 Ann. Math. 190 249–305).

Sharp well-posedness and ill-posedness results for dissipative KdV equations on the real line

Sharp well-posedness and ill-posedness results for dissipative KdV equations on the real line

Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

<p style='text-indent:20px;'>We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in the anisotropic Sobolev spaces <inline-formula><tex-math id="M1">\begin{document}$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M2">\begin{document}$ \beta <0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \gamma >0, $\end{document}</tex-math></inline-formula> we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M4">\begin{document}$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ s_{1}>-\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ s_{2}\geq 0 $\end{document}</tex-math></inline-formula>. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in <inline-formula><tex-math id="M7">\begin{document}$ H^{s_{1},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ s_{1}<-\frac{1}{2} $\end{document}</tex-math></inline-formula> in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not <inline-formula><tex-math id="M9">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M10">\begin{document}$ \beta <0,\gamma >0, $\end{document}</tex-math></inline-formula> by using the <inline-formula><tex-math id="M11">\begin{document}$ U^{p} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ V^{p} $\end{document}</tex-math></inline-formula> spaces, we prove that the Cauchy problem is locally well-posed in <inline-formula><tex-math id="M13">\begin{document}$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula>.</p>

Sharp well-posedness for the generalized KdV of order three on the half line

In this paper we study the generalized Korteweg–de Vries (KdV) equation with the nonlinear term of order three: (u3+1)x. We prove sharp local well-posedness for the initial and boundary value problem posed on the right half line. We thus close the gap in the well-posedness theory of the generalized KdV which remained open after the seminal work of Colliander and Kenig in Colliander and Kenig (2002).

Sharp well-posedness for a coupled system of mKdV-type equations

We consider the initial value problem associated with a system consisting modified Korteweg–de Vries-type equations $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv + \partial _x^3v + \partial _x(vw^2) =0,&{}\quad u(x,0)=\phi \partial _tw + \alpha \partial _x^3w + \partial _x(v^2w) =0,&{}\quad v(x,0)=\psi (x), \end{array}\right. } \end{aligned}$$and prove the local well-posedness results for given data in low regularity Sobolev spaces $$H^{s}({\mathbb {R}})\times H^{s}({\mathbb {R}})$$, $$s> -\,\frac{1}{2}$$, for $$0 1$$ as well.

Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier–Besov spaces

We study the well-posedness and ill-posedness for the Cauchy problem of the three-dimensional primitive equations describing the large-scale oceanic and atmospheric circulations. By using the Littlewood–Paley analysis technique, we prove that the Cauchy problem of the three-dimensional primitive equations with the Prandtl number P=1 is locally well-posed in the Fourier–Besov spaces FḂp,r2−3p(R3) for 1<p≤∞,1≤r<∞ and FḂ1,r−1(R3) for 1≤r≤2, and is globally well-posed in these spaces when the initial data are small. We also verify that such problem is ill-posed in FḂ1,r−1(R3) for 2<r≤∞, which implies that our work completes a dichotomy of well-posedness and ill-posedness for the three-dimensional primitive equations in the Fourier–Besov space framework.

Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{R}^+ $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrodinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the \begin{document}$L^2$\end{document} conservation law of the Schrodinger part with a careful iteration of the rough wave part in lower order Sobolev norms in the spirit of the work in [ 5 ].

Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier–Besov spaces

We study the Cauchy problem of the incompressible micropolar fluid system in R3. In a recent work of the first author and Jihong Zhao (Zhu and Zhao, 2018), it is proved that the Cauchy problem of the incompressible micropolar fluid system is locally well-posed in the Fourier–Besov spaces FḂp,r2−3p for 1<p≤∞ and 1≤r<∞, and globally well-posed in these spaces with small initial data. In this work we consider the critical case p=1. We show that this problem is locally well-posed in FḂ1,r−1 for 1≤r≤2, and is globally well-posed in these spaces with small initial data. Furthermore, we prove that such problem is ill-posed in FḂ1,r−1 for 2<r≤∞, which implies that the function space FḂ1,2−1 is sharp for well-posedness. In addition, using a similar argument we also prove that this problem is ill-posed in the Besov space Ḃ∞,r−1 with 2<r≤∞.

Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

In this paper, we investigate the Cauchy problem for the fourth order nonlinear Schrodinger equation \begin{document}$i \partial_{t}u+\partial_{x}^{4}u=u^{2}, (t,x)∈[0,T]× \mathbb{R}.$ \end{document} Zheng (Adv. Differential Equations, 16(2011), 467-486.) has proved that the problem is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} with \begin{document}$-\frac{7}{4} In this paper, we aim at extending Zheng's work to a lower regularity index. We prove that the equation is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} when \begin{document}$s≥q -2$\end{document} and ill-posed when \begin{document}$s in the sense that the solution map is discontinuous for \begin{document}$s . The key ingredient used in this paper is Besov-type space introduced by Bejenaru and Tao (Journal of Functional Analysis, 233(2006), 228-259.).

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

The goal of this paper is two-fold. Firstly, by using the Fourier restriction norm method and the fixed point theorem, we prove that the Cauchy problem for a generalized Ostrovsky equation \t\t\t∂x(ut−β∂x3u+13∂x(u3))−γu=0,β>0,γ>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\partial_{x} \\biggl(u_{t}-\\beta\\partial_{x}^{3}u +\\frac{1}{3}\\partial_{x}\\bigl(u^{3}\\bigr) \\biggr) - \\gamma u=0,\\quad\\beta>0,\\gamma>0, $$\\end{document} is locally well-posed in H^{s}(mathbf{R}) with sgeqfrac{1}{4}. Secondly, we prove that the Cauchy problem for a generalized Ostrovsky equation is not well-posed in H^{s}(mathbf{R}) with s<frac{1}{4} in the sense that the solution map is C^{3}.

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation ut+∂x2n+1u=∂x(u∂xnu)+∂xn-1(ux2), n≥2,n∈N+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H(-n2+34,-12n)(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H(s,a)(R) with s<-n2+34 and all a∈R.

Sharp well-posedness for the Chen-Lee equation

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(\mathbb{R})$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.

On the Market-Neutrality of Optimal Pairs-Trading Strategies

We consider the problem of optimal investment in a market with two cointegrated stocks and an agent with CRRA utility. We extend the findings of Liu and Timmermann [The Review of Financial Studies, 26(4):1048-1086, 2013] by paying special attention to when/if the associated stochastic control problem is well-posed and providing a verification result. Our new findings lead to a sharp well-posedness condition which is, surprisingly, also the necessary and sufficient condition for the optimal investment to be market-neutral (i.e. having offsetting long/short positions in the stocks). Hence, we provide a theoretical justification for market-neutral pairs-trading which, despite having a strong practical relevance, has been lacking a theoretical ground.

Sharp Well-Posedness and Ill-Posedness for the Chern–Simons–Dirac System in One Dimension

Sharp Well-Posedness and Ill-Posedness for the Chern–Simons–Dirac System in One Dimension

Sharp well-posedness for the Benjamin equation

Having the ill-posedness in the range s < − 3 / 4 of the Cauchy problem for the Benjamin equation with an initial H s ( R ) data, we prove that the already-established local well-posedness in the range s > − 3 / 4 of this initial value problem is extendable to s = − 3 / 4 and also that such a well-posed property is globally valid for s ∈ [ − 3 / 4 , ∞ ) .