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Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces P Λ described by lower sets Λ . Given a budget n for the dimension of P Λ , we prove that certain lower sets Λ n , with cardinality n , provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case d = ∞ . We also emphasize obtaining results which hold for the full range n ≥ 1 , rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

In this paper, we study the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interactions and initial data belonging to a class of analytic Gevrey functions. Here, we present a local well-posedness result in the analytic Gevrey class G σ , s × G σ , s by proving some bilinear estimates in Bourgain’s space with exponential weight. Furthermore, we prove that the obtained solution can be extended to any time T > 0 , as long as the radius of the spatial analyticity σ is bounded below by c T - 2 , if 0 < a < 1 / 2 , or c T - 4 , if a > 1 / 2 .

We consider the initial value problems (IVPs) for the modified Korteweg–de Vries (mKdV) equation ∂ t u + ∂ x 3 u + μ u 2 ∂ x u = 0 , x ∈ R , t ∈ R , u ( x , 0 ) = u 0 ( x ) , where u is a real valued function and μ = ± 1 , and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short) ∂ t v + i α ∂ x 2 v + β ∂ x 3 v + i γ | v | 2 v = 0 , x ∈ R , t ∈ R , v ( x , 0 ) = v 0 ( x ) , where α , β and γ are real constants and v is a complex valued function. In both problems, the initial data u 0 and v 0 are analytic on R and have uniform radius of analyticity σ 0 in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same σ 0 till some lifespan 0 < T 0 ≤ 1 . We also consider the evolution of the radius of spatial analyticity σ ( t ) when the local solution extends globally in time and prove that for any time T ≥ T 0 it is bounded from below by c T - 4 3 , for the mKdV equation in the defocusing case ( μ = - 1 ) and by c T - ( 4 + ε ) , ε > 0 , for the tNLS equation. The result for the mKdV equation improves the one obtained in Bona et al. (Ann Inst Henri Poincaré 22:783–797, 2005) and, as far as we know, the result for the tNLS equation is the new one.

We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

In this paper, we study the qualitative behavior of the Cauchy problem of a hyperbolic model $\\left\\{ \\matrix{p_t} - \\nabla \\cdot \\left( {pq} \\right)\\, = \\,\\Delta p,\\,x \\in {{\\Cal R}^d},t > 0, \\hfill \\cr{q_t} - \\nabla \\left( {\\varepsilon |q{|^2} + p} \\right)\\, = \\,\\varepsilon \\Delta q,\\hfill \\cr\\endmatrix \\right.$ which is transformed from a singular chemotaxis system describing the effect of a reinforced random walk in [17,27]. When d = 1 and the initial data are prescribed around a constant ground state (p̄,0) with p̄≥ 0, we prove the global asymptotic stability of constant ground states, and identify the explicit decay rate of solutions under very mild conditions on initial data. Moreover, we study the diffusion (viscous) limit of solutions as ɛ ➞ 0 with convergence rates toward solutions of the non-diffusible (inviscid) problem. While the existence of global large solutions of the system in multi-dimensions remains an outstanding open question, we show that the model exhibits a strong parabolic smoothing effect: namely, solutions are spatially analytic for a short time provided that the initial data belong to Lq(ℝd) for any q > d ≥ 1. In fact, when d = 1, we obtain that the solution remains real analytic for all time.

The main focus of this article is to investigate the generalized fractional drift-diffusion system with small initial data in Besov-Morrey spaces. Our goal is to establish the global well-posedness and asymptotic stability of mild solutions for this system. The results obtained in this study have broad applicability in the modeling of various types of fractional parabolic systems. In other words, the techniques developed here can be useful in studying similar types of systems in the future.

This paper is devoted to obtaining new lower bounds to the radius of spatial analyticity for the solutions of modified Korteweg–de Vries (mKdV) equation and a coupled system of mKdV-type equations, starting with real analytic initial data with a fixed radius of analyticity σ 0 . Specifically, we derive almost conserved quantities to prove that the local solution can be extended to a time interval [0,  T ] for any large T > 0 in such a way that the radius of analyticity σ ( T ) decays no faster than c T - 1 for both the equations, where c is a positive constant. The results of this paper improve the ones obtained in Figueira and Panthee (Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion, to appear in NoDEA, arXiv:2307.09096 ) and Figueira and Himonas (J Math Anal Appl 497(2):124917, 2021), respectively, for the mKdV equation and a mKdV-type system.

The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

We prove that the profile of a periodic traveling wave propagating at the surface of water above a flat bed in a flow with a real analytic vorticity must be real analytic, provided the wave speed exceeds the horizontal fluid velocity throughout the flow. The real analyticity of each streamline beneath the free surface holds even if the vorticity is only Hölder continuously differentiable.