Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such solutions have been proved to exist in Ionescu– Pusateri [Invent. Math. 199 (2015)], Alazard–Delort [Ann. Sci. Éc. Norm. Supér. (4) 48 (2015)], Hunter et al. [Comm. Math. Phys. 346 (2016)], and Ifrim–Tataru [Bull. Soc. Math. France 144 (2016)] in much higher regularity. Our goal in this paper is to improve these results and prove global well-posedness under minimal regularity and decay assumptions for the initial data. One key ingredient here is represented by the balanced cubic estimates in our first paper. Another is the nonlinear vector field Sobolev inequalities, an idea first introduced by Ifrim–Tataru [Ann. Sci. Éc. Norm. Supér. (4) 52 (2019)] in the context of the Benjamin–Ono equations.

The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a sharp local well-posedness result. The second goal of this article is to consider the long-time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: “Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions”; the conjecture was initially proved for a class of semilinear Schrödinger type models. Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has phase rotation symmetry and is conservative and defocusing, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows where no localization condition is imposed on the data. Furthermore, we prove the global well-posedness at the minimal Sobolev regularity as in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that small, $\epsilon $ ϵ size data yields long-time solutions on the $\epsilon ^{-8}$ ϵ − 8 time-scale. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows, also at minimal regularity.

We study scattering theory for the semilinear wave equation Utt Au = JujP-lu in two space dimensions. We show that if p > p0 = (3+x/ii)/2, the scattering operator exists for smooth and small data. The lower bound p0 of p is considered to be optimal (see Glassey [6, 7], Schaeffer [ 18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at t = 0 in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: u(x, t) = u(x, t)? I[ [(II )(,s dy ds, ? 27r IJ Jix-yl<t-s /(tS)2 Ix y12 for t e R, where u-(x, t) is a solution of utt Au = 0 which u(x, t) approaches asymptotically as t -+ -oo. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of utt Au = Iu Pu in two space dimensions.

We deal with systems of quasilinear wave equations which contain quadratic nonlinearities in 2-dimensional space. We have already known that such the system has a smooth solution till the time t0 = Ce-2 for sufficiently small e > 0, where e is the size of initial data. In this paper, we shall show that if quadratic and cubic nonlinearities satisfy so-called Null-condition, then the smooth solution exists globally in time. In the proof of the theorem, we use the Alinhac ghost weight energy.

We study the generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation in dimensions and with nonlinearity of degree , a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if . Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [], generalize those obtained in [] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

Let T denote a closed oriented surface and let there be given a basis α1,..., αn, β1,..., βn of H1 (T; ℤ) with αi · αj = βi · βj = 0, αi · βj = δij as intersection numbers. Then one can construct an ordinary imbedding of T in 3-dimensional euklidian space, such that the given basis is represented by the meridians and parallels of latitude of that imbedding. If there is a given imbedding of T into an euclidian space of dimension n≥5, then one has a factorisation through an ordinary imbedding into 3-dimensional space, such that the given basis is represented by the meridians and parallels of latitude of that ordinary imbedding. If in the case n=4 the imbedding of T can be factorised through 3-dimensional space one has a further invariant. Besides the skewsymmetric intersection form there is a quadratic form which must take its normal form on the given basis in order to represent this basis by meridians and parallels of latitude as above.

We prove uniqueness and continuous dependence on initial data of weak solutions of the Navier--Stokes equations of compressible flow in two and three space dimensions. The solutions we consider may display codimension-one discontinuities in density, pressure, and velocity gradient, and consequently are the generic singular solutions of this system. The key point of the analysis is that solutions with minimal regularity are best compared in a Lagrangean framework; that is, we compare the instantaneous states of corresponding fluid particles in two different solutions rather than the states of different fluid particles instantaneously occupying the same point of space-time. Estimates for $H^{-1}$ differences in densities and $L^2$ differences in velocities are obtained by duality from bounds for the corresponding adjoint system.

This note examines the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for three-dimensional nonlinear wave equations of the form [partial derivative][sub t][sup 2]u [minus] [Delta]u = G(u, Du), (1), u(0) = f, [partial derivative][sub t]u(0) = g, (2), where Du = ([partial derivative][sub t]u, [del][sub x]u). For nonintegral s, this requires the use of an inequality. Thus it is crucial to gain control of the L[sup [ell][minus]1] norm in time of the maximum norm of the gradient of the solution in space. This is usually done by using the Sobolev imbedding theorem which leads to the restriction s > n/2+1 for the Sobolev exponent. The authors show that in three space dimensions (the case to which they restrict themselves throughout the paper), the lower bound for the Sobolev exponent can be reduced from 5/2 to s([ell]) [identical to] max[l brace]2, (5[ell] [minus] 7)/(2[ell] [minus] 2)[r brace] when the nonlinearity G in (1) grows no faster than order [ell] in Du. Thus, as [ell] [yields] [infinity] the classical result s > 5/2 is approached. They also show that this is sharp in the sense that the quantity [parallel]f[parallel][sub H[sup s([ell])]] + [parallel]g[parallel][sub H[supmore » s([ell])[minus]1]] is not sufficient, in general, to control the local existence time of solutions (for [ell] [ge] 3). The authors return to the spherically symmetric case at the end of the paper. For general nonlinearities, such a result just fails. It is shown how to apply instead a space-time estimate for the free solution due to Marshall and Pecher, as an extension of the work of Strichartz. 9 refs.« less

We consider the initial value problem for systems of nonlinear Klein–Gordon equations with quadratic nonlinearities. We prove the existence of scattering states, namely, the asymptotic stability of small solutions in the neighborhood of the free solutions for small initial data in the weighted Sobolev space H4,3(R3)×H3,3(R3). If nonlinearities satisfy the strong null condition, then the same result is true in two space dimensions for small initial data in H5,4(R2)×H4,4(R2). A system of massive Dirac–massless Klein–Gordon equations in three space dimensions is also considered by our method.

We study scattering theory for the semilinear wave equation u t t − Δ u = | u | p − 1 u {u_{tt}} - \Delta u = |u{|^{p - 1}}u in two space dimensions. We show that if p &gt; p 0 = ( 3 + 17 ) / 2 p &gt; {p_0} = (3 + \sqrt {17} )/2 , the scattering operator exists for smooth and small data. The lower bound p 0 {p_0} of p is considered to be optimal (see Glassey [6, 7], Schaeffer [18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at t = 0 t = 0 in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: \[ u ( x , t ) = u 0 − ( x , t ) + 1 2 π ∫ − ∞ t ∫ | x − y | ≤ t − s ( | u | p − 1 u ) ( y , s ) ( t − s ) 2 − | x − y | 2 d y d s , u(x,t) = u_0^ - (x,t) + \frac {1}{{2\pi }}\int _{ - \infty }^t {\int _{|x - y| \leq t - s} {\frac {{(|u{|^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - |x - y{|^2}} }}dy\;ds,} } \] for t ∈ R t \in R , where u 0 − ( x , t ) u_0^ - (x,t) is a solution of u t t − Δ u = 0 {u_{tt}} - \Delta u = 0 which u ( x , t ) u(x,t) approaches asymptotically as t → − ∞ t \to - \infty . The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of u t t − Δ u = | u | p − 1 u {u_{tt}} - \Delta u = |u{|^{p - 1}}u in two space dimensions.

We study scattering theory for the semilinear wave equation ${u_{tt}} - \Delta u = |u{|^{p - 1}}u$ in two space dimensions. We show that if $p > {p_0} = (3 + \sqrt {17} )/2$, the scattering operator exists for smooth and small data. The lower bound ${p_0}$ of p is considered to be optimal (see Glassey [6, 7], Schaeffer [18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at $t = 0$ in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: \[ u(x,t) = u_0^ - (x,t) + \frac {1}{{2\pi }}\int _{ - \infty }^t {\int _{|x - y| \leq t - s} {\frac {{(|u{|^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - |x - y{|^2}} }}dy\;ds,} } \] for $t \in R$, where $u_0^ - (x,t)$ is a solution of ${u_{tt}} - \Delta u = 0$ which $u(x,t)$ approaches asymptotically as $t \to - \infty$. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of ${u_{tt}} - \Delta u = |u{|^{p - 1}}u$ in two space dimensions.

We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost for localized initial data and, in the dispersive case, for initial data which are semiclassically microlocalized.

A vector field method for some nonlinear Dirac models in Minkowski spacetime

This article concerns the global-in-time existence of smooth solutions with small amplitude to two space dimensional Euler-Poisson system. The main difficulty lies in the slow time decay (1 + t)−1 of the linear system. Inspired by the work of Ozawa et al., [“Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions,” Math. Z. 222, 341–362 (1996)10.1007/BF02621870; “Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions,” in Nonlinear Waves, Gakuto International Series: Mathematical Sciences and Applications Vol. 10 (Gakkotosho, Tokyo, 1997), pp. 383–392,] we show that such smooth solutions exist for radially symmetric flows.

We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time T N → 0 T_N\to 0 and solution u N ( t ) u_N(t) such that u N ( T N ) → ∞ u_N(T_N)\to \infty in the Besov space B 2 , σ − 1 ( R ) B_{2,\sigma }^{-1}(\mathbb {R}) ( σ &gt; 2 \sigma &gt;2 ) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space H − 1 ( R 2 ) H^{-1}(\mathbb {R}^2) . We systematically utilize the modulation space M 2 , 1 0 M_{2,1}^0 for one dimension and the scaled modulation space ( M 2 , 1 0 ) N (M_{2,1}^0)_N for two dimensions.