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In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice \mathbb {Z}. The level sets of solutions with step-like initial conditions are located at position c_*t-\frac {3}{2\lambda _*}\ln t+O(1) as t\rightarrow +\infty for some explicit positive constants c_* and \lambda _*. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed c_* uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times. [ABSTRACT FROM AUTHOR]

Constrained gradient flows are studied in fracture mechanics to describe strongly irreversible (or unidirectional) evolution of cracks. The present paper is devoted to a study on the long-time behavior of non-compact orbits of such constrained gradient flows. More precisely, traveling wave dynamics for a one-dimensional fully nonlinear Allen-Cahn type equation involving the positive-part function is considered. Main results of the paper consist of a construction of a one-parameter family of degenerate traveling wave solutions (even identified when coinciding up to translation) and exponential stability of such traveling wave solutions with some basin of attraction, although they are unstable in a usual sense. [ABSTRACT FROM AUTHOR]

This paper is concerned with the algebraic and analytic q-de Rham complexes attached to linear q-difference operators with Laurent polynomial coefficients over the field of complex numbers. There is a natural morphism from the former to the latter complex. Whether or not it is a quasi-isomorphism, i.e. , whether or not the induced morphisms on the corresponding cohomology spaces are isomorphisms, is the basic question considered in the present paper. We study this question following three distinct approaches. The first one is based on duality, and leads to a direct connection between the problem considered in the present paper and the convergence of formal series solutions of q-difference equations. The second approach is sheaf theoretic, based on growth considerations. The third one relies on the local analytic theory of q-difference equations. The paper ends with an extension of our results to variants of the above q-de Rham complexes when certain q-spirals of poles are allowed. Our study includes the case |q| = 1. [ABSTRACT FROM AUTHOR]

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and local functions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with N-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating N point vortices and counter-rotating pairs. Some other quantitative properties are also established. [ABSTRACT FROM AUTHOR]

This paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochastic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup. [ABSTRACT FROM AUTHOR]

The Hessian quotient equations \begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation} were studied for k-th symmetric elementary function S_k(D^2u) of eigenvalues \lambda (D^2u) of the Hessian matrix D^2u, where 0\leq l

In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by the Christoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring's problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in ⋅ C1(Rd) ∩ L2(Rd). [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following equations {(−Δ)m+α/2u(x) = ƒ(x,u,Du,⋅⋅⋅), x ∈ Rn, u ∈ C2m+[α],{α}+εloc∩ Lα(Rn), u(x) ≥ 0, x ∈ Rn involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to the above equations. Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to the above fractional higher-order equations with general nonlinearities ƒ(x,u,Du,⋅⋅⋅) including conformally invariant and odd order cases. In particular, we classify nonnegative classical solutions to all odd order conformally invariant equations. Our results completely improve the classification results for third order conformally invariant equations in Dai and Qin (Adv. Math., 328 (2018), 822-857) by removing the assumptions on integrability. We also give a crucial characterization for \alpha-harmonic functions via outer-spherical averages in the appendix. [ABSTRACT FROM AUTHOR]

In this paper, we develop an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. As an application of the theorem, we study the quintic nonlinear Schrödinger equation on the two-dimensional torus iut-Δ u + |u|4u = 0, x ∈ T2 := R2/(2π Z)2, t ∈ R. We obtain a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The idea in our proof comes from Geng, Xu, and You [Adv. Math. 226 (2011), pp. 5361–5402], which however has to be substantially developed to deal with the equation above. [ABSTRACT FROM AUTHOR]

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a δ-subharmonic function is established allowing the case of hyper-order equal to one and minimal hyper-type, which improves the condition of the hyper-order less than one. Finally, we make a careful discussion of a well-known difference equation and give the possible forms of the equation under a growth condition for the solutions. [ABSTRACT FROM AUTHOR]

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191-7223, is that the structure algebra K[M(X,r)], over a field K, of a finite bijective left non-degenerate solution (X,r) of the Yang-Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M(X,r) is central-by-finite. This however is not true, even in case (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M(X,r) is abelian-by-finite and thus the conclusion that K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution (X,r) is square-free is omitted. [ABSTRACT FROM AUTHOR]

In this paper we study axially symmetric solutions of the Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between 4 and 10. In dimension 3, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals 1. [ABSTRACT FROM AUTHOR]

There exists a holomorphic quadratic differential defined on any H-surface immersed in the homogeneous space E(κ t) given by U. Abresch and H. Rosenberg, called the Abresch-Rosenberg differential. However, there was no Codazzi pair on such an H-surface associated with the Abresch-Rosenberg differential when t ≠ 0. The goal of this paper is to find a geometric Codazzi pair defined on any H-surface in E(κ, τ), when t ≠ 0, whose (2, 0)-part is the Abresch-Rosenberg differential. We denote such a pair as (I, IIAR), were I is the usual first fundamental form of the surface and IIAR is the Abresch-Rosenberg second fundamental form. In particular, this allows us to compute a Simons' type equation for H-surfaces in E(κ, τ). We apply such Simons' type equation, first, to study the behavior of complete H-surfaces S of finite Abresch-Rosenberg total curvature immersed in E(κ, τ). Second, we estimate the first eigenvalue of any Schrodinger operator L = Δ+V, V continuous, defined on such surfaces. Finally, together with the Omori-Yau maximum principle, we classify complete H-surfaces in E(κ, τ), τ ≠ 0, satisfying a lower bound on H depending on κ, τ, and an upper bound on the norm of the traceless IIAR, a gap theorem. [ABSTRACT FROM AUTHOR]

We prove that, for every invertible horizontal-like map (i.e., Hénon-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the d, \bar \partial, and dd^c equations on convex domains. [ABSTRACT FROM AUTHOR]

The known logarithmic-type modulus of continuity is improved for weak solutions to the two-phase Stefan problem in low space dimensions. In the two dimensional problem, the modulus becomes \boldsymbol \omega (r)\approx \operatorname {exp}\{-c|\operatorname {ln} r|^{\frac 12}\}, for some c\in (0,1). In the one dimensional problem, the modulus becomes Hölder-type. The main merit lies in a delicate, potential theoretical estimate that sharpens the expansion of positivity for non-negative super-solutions to parabolic equations. Our argument offers a unified approach to the state-of-the-art moduli in all dimensions. [ABSTRACT FROM AUTHOR]

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W^{1,1} solutions are necessarily W^{1,2}_{\operatorname {loc}}, which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl's classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W^{1,1} solution to be improved to W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution. [ABSTRACT FROM AUTHOR]

In this paper, we give a resolution of the generalized Fermat equations x5 + y5 = 3zn and x13 + y13 = 3zn, for all integers n ≥ 2 and all integers n ≥ 2 which are not a power of 7, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents n. We also give a number of results for the equations x5 + y5 = dzn, where d = 1, 2, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat’s Last Theorem and which uses a new application of level raising at p modulo p. [ABSTRACT FROM AUTHOR]

The aim of the paper is to state removable singularities results for solutions of fully nonlinear degenerate elliptic equations without any knowledge of the behaviour of the solution approaching the singular set and to obtain unconditional results of Brezis-Veron type for operators defined as the partial sum of the eigenvalues of the Hessian matrix. [ABSTRACT FROM AUTHOR]

In this paper, we develop an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. As an application of the theorem, we study the quintic nonlinear Schrödinger equation on the two-dimensional torus iut-Δ u + |u|4u = 0, x ∈ T2 := R2/(2π Z)2, t ∈ R. We obtain a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The idea in our proof comes from Geng, Xu, and You [Adv. Math. 226 (2011), pp. 5361–5402], which however has to be substantially developed to deal with the equation above. [ABSTRACT FROM AUTHOR]

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space. [ABSTRACT FROM AUTHOR]

The global asymptotic dynamics of point vortices for the lake equations is rigorously derived. Vorticity that is initially sharply concentrated around N distinct vortex centers is proven to remain concentrated for all times. Specifically, we prove weak concentration of the vorticity and in addition strong concentration in the direction of the steepest ascent of the depth function. As a consequence, we obtain the motion law of point vortices following at leading order the level lines of the depth function. The lack of strong localization in the second direction is linked to the vortex filamentation phenomena. The main result allows for any fixed number of vortices and general assumptions on the concentration property of the initial data to be considered. No further properties such as a specific profile or symmetry of the data are required. Vanishing topographies on the boundary are included in our analysis. Our method is inspired by recent results on the evolution of vortex rings in 3D axisymmetric incompressible fluids. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the local low regularity of solutions and decay estimates of solitary waves to the rotation-modified Kadomtsev-Petviashvili (rmKP) equation. It is shown that with negative dispersion, the rmKP equation is locally well-posed for data in Hs1,s2 (R2) for s1 > - 3/10 and s2 ≥ 0, and hence globally well-posed in the space L2. Moreover, an improved result on the decay property of the solitary waves is established, which shows that all solitary waves of the rmKP equation decay exponentially at infinity. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the \operatorname {SL}_2(\mathbb {C})–character scheme of a finitely generated group. Given a presentation of a finitely generated group \Gamma, we give equations defining the coordinate ring of the scheme of \operatorname {SL}_2(\mathbb {C})–characters of \Gamma (finitely many equations when \Gamma is finitely presented). We also study the scheme of abelian and non-simple representations and characters. Finally we apply our results to study the \operatorname {SL}_2(\mathbb {C})–character scheme of the Borromean rings. [ABSTRACT FROM AUTHOR]

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation \begin{equation*} \partial_t u + u \partial_x u + \partial_x^3 u = f \tag{i}\label{star} \end{equation*} on the interval $0\leq x\leq 2\pi , t\geq 0 $, with periodic boundary conditions \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag{ii}\label{2star} \end{equation*} where the distributed control $f\equiv f(x,t)$ is restricted so that the ``volume'' $\int ^{2\pi }_0 u(x,t) dx $ of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of {\em open loop } control, if the control $f$ is allowed to act on the whole spatial domain $(0,2\pi)$, it is shown that the system is globally exactly controllable, i.e., for given $T> 0$ and functions $\phi (x)$, $\psi (x)$ with the same ``volume'', one can alway find a control $f$ so that the system (i)--(ii) has a solution $u(x,t)$ satisfying [ u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi(x) .] If the control $f$ is allowed to act on only a small subset of the domain $(0,2\pi)$, then the same result still holds if the initial and terminal states, $\psi $ and $\phi $, have small ``amplitude'' in a certain sense. In the case of {\em closed loop} control, the distributed control $f$ is assumed to be generated by a linear feedback law conserving the ``volume'' while monotonically reducing $\int ^{2\pi}_0 u(x,t)^2 dx $. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper. [ABSTRACT FROM AUTHOR]

We develop a variety of new techniques to treat Diophantine equations of the shape x^2+D =y^n, based upon bounds for linear forms in p-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers x and y, and n \geq 3, with the property that y^n > x^2 and x^2-y^n has no prime divisor exceeding 11. [ABSTRACT FROM AUTHOR]

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space \mathbb {R}^3, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in \mathbb {R}^4, to the energy-subcritical equation in \mathbb {R}^3. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. [ABSTRACT FROM AUTHOR]

Given a hyperelliptic curve C of genus g over a number field K and a Weierstrass model \mathscr {C} of C over the ring of integers \mathcal {O}_K (i.e. the hyperelliptic involution of C extends to \mathscr {C} and the quotient is a smooth model of \mathbb {P}^1_K over \mathcal {O}_K), we give necessary and sometimes sufficient conditions for \mathscr {C} to be defined by a global Weierstrass equation. In particular, if C has everywhere good reduction, we prove that it is defined by a global integral Weierstrass equation with invertible discriminant if the class number h_K is prime to 2(2g+1), confirming a conjecture of M. Sadek. [ABSTRACT FROM AUTHOR]

For dimensions n \geq 3, we classify singular solutions to the generalized Liouville equation (-\Delta)^{n/2} u = e^{nu} on \mathbb R^n \setminus \{0\} with the finite integral condition \int _{\mathbb {R}^n} e^{nu} < \infty in terms of their behavior at 0 and \infty. These solutions correspond to metrics of constant Q-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser–Trudinger inequality for radial functions. [ABSTRACT FROM AUTHOR]

We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying as |x| \to +\infty, which leads to ambiguity in the velocity \vec {R}^\perp \theta. This ambiguity is corrected by imposing m-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential non-uniqueness, due to symmetry-breaking bifurcations, in analogy with work (see Guillod and Sverák [ Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces , 2017], and Jia and Sverák [J. Funct. Anal. 268 (2015), pp. 3734–3766]) on the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

The article presents a study to examine the solvability of second-order equations with partially BMO coefficients. It analyses the Lp-solvability of second-order elliptic and parabolic equations in nondivergence form using divergence-form equations. It mentions the leading coefficients has been assumed to be in locally BMO spaces.

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the $W^{1,p} (1

This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called \textit{arithmetic non-degeneracy}. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right} $ called the \textit{arithmetic Newton polygon} of $f(x,y)$, and introduce the notion of \textit{arithmetic non-degeneracy with respect to }$\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$. [ABSTRACT FROM AUTHOR]

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain $D$, where its integrals over a family of subdomains in $D$ are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when $D$ is a bounded domain in ${\mathbb{R}}^2$ with a piecewise smooth boundary. Some intermediate results related to dynamical systems with {\it two} generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain $D\subset {\mathbb{R}}^2$ with data on the {\it whole} boundary $\partial D$. [ABSTRACT FROM AUTHOR]

In this paper we study the equation \[ u_t=\Delta(\phi(u) - \lambda f(u) + \lambda u_t) + f(u) \] in a bounded domain of $\mathbb{R}^d$, $d\ge1$, with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by $\phi$, and total birth and mortality rates characterized by $f$. We will show that the aggregating mechanism induced by $\phi(u)$ allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function $\phi(u)$. [ABSTRACT FROM AUTHOR]

We investigate a regularity for weak solutions of the following generalized Leray equations (−Δ)α V − 2α−1/2α V + V ⋅ ∇ V − 1/2α x ⋅ ∇V + ∇P = 0, which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in R3. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions V in the weighted Hilbert space Hαω(R3). Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from Hαω(R3) to Hω1+α(R3). This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of V which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44. [ABSTRACT FROM AUTHOR]

Using an extension of a recent method of Cabré and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabré and Martel from bounded smooth domains to unbounded nonsmooth domains. [ABSTRACT FROM AUTHOR]

In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family $L_{\xi_0}$ of left invariant operators on a free nilpotent Lie group. The fundamental solution $\Gamma_{\xi_0}$ of the operator $L_{\xi_0}$ is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is $C^\infty$. [ABSTRACT FROM AUTHOR]

In this paper we introduce recursive functions \begin{align*} &{}^{\mathbf{Fi}}(x_1,x_2)^{\lambda_1,\dotsc,\lambda_s}\qquad(s\ge0), &{}^{\mathbf{Th}}(x_1,x_2,x_3)_i^{\lambda_1,\dotsc,\lambda_{2s}} \qquad(i=1,2,3;s\ge0), &{}^{\mathbf{Ro}}(x_1,x_2,x_3)_i^{\mu_1,\dotsc,\mu_s}\qquad(i=1,2,3;s\ge0) \end{align*} of the word variables $x_1,x_2,x_3$, natural number variables $\lambda_k$ and variables $\mu_k$ whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation \[x_1x_2x_3x_4=\zeta(x_1,x_2,x_3)x_5,] where $\zeta(x_1,x_2,x_3)$ is an arbitrary word in the alphabet $x_1,x_2,x_3$, in a free monoid. [ABSTRACT FROM AUTHOR]

For the generalized surface quasi-geostrophic equation { ∂tθ +u ⋅ ∇ θ = 0, in R2 × (0,T), u = ∇⊥ ψ, ψ = (−Δ)−s θ in R2 × (0,T), 0 < s < 1, we consider for k ≥ 1 the problem of finding a family of k-vortex solutions θε (x,t) such that as ε → 0 { θε (x,t) → ∑j=1k mjδ (x-ξj(t) for suitable trajectories for the vortices x = ξj(t). We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem { (−Δ)sW = (W−1)γ+, in R2, 1 < γ < 1+s}/1−s whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42]. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space Rd+, where the coefficients are the product of xdα, α ∈ (−∞, 1), and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary {xd = 0} and they may not be locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations. [ABSTRACT FROM AUTHOR]

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a "triangle". [ABSTRACT FROM AUTHOR]

We study a stochastic Schrödinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension d ≤ 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schrödinger group, which allows us to cope with the strong irregularity of the solution. [ABSTRACT FROM AUTHOR]

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular j function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on j with algebraic coefficients. [ABSTRACT FROM AUTHOR]

We study here the Zakharov-Kuznetsov equation in dimension 2, 3 and 4 and the modified Zakharov-Kuznetsov equation in dimension 2. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of K solitons Rk (with distinct velocities), we prove the existence and uniqueness of a multi-soliton u such that |u(t) − ∑k=1K Rk(t)|H1 → 0 as t → + ∞. The convergence takes place in Hs with an exponential rate for all s ≥ 0. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of H1-norms of the errors (inspired by Martel [Amer. J. Math. 127 (2005), pp. 1103-1140]), and introduce a new ingredient for the control of the Hs-norm in dimension d ≥ 2, by a technique close to monotonicity. [ABSTRACT FROM AUTHOR]

We develop Green's function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. [ABSTRACT FROM AUTHOR]

We are concerned with a super-Liouville equation on compact surfaces with genus larger than one, obtaining the first non-trivial existence result for this class of problems via min-max methods. In particular we make use of a Nehari manifold and, after showing the validity of the Palais-Smale condition, we exhibit either a mountain pass or linking geometry. [ABSTRACT FROM AUTHOR]

Let X be the family of hypersurfaces in the odd-dimensional torus T2n+1 defined by a Laurent polynomial ƒ with fixed exponents and variable coefficients. We show that if n Δ, the dilation of the Newton polytope Δ of ƒ by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n(X) has a full set of algebraic solutions (where W• denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations. [ABSTRACT FROM AUTHOR]