We consider a system of $n$-th order nonlinear quasilinear partial differential equations of the form
$${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left( {\bf x}, t, \{\partial_{\bf x}^{\bf j} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) = {\bf {u}}_I({\bf x})$$
with $\mathbf{u}\in\CC^{r}$, for $ t\in (0,T)$ and large $|{\bf x}|$ in a poly-sector $S$ in $\mathbb{C}^d$ ($\partial_{\bf x}^{\bf j} \equiv \partial_{x_1}^{j_1} \partial_{x_2}^{j_2} ...\partial_{x_d}^{j_d}$ and $j_1+...+j_d\le n$). The principal part of the constant coefficient $n$-th order differential operator $\mathcal{P}$ is subject to a cone condition. The nonlinearity ${\bf g}$ and the functions $\mb u_I$ and $\mb u$ satisfy analyticity and decay assumptions in $S$.The paper shows existence and uniqueness of the solution of this problem and finds its asymptotic behavior for large $|\bf x|$. Under further regularity conditions on $\mb g$ and $\mb u_I$ which ensure the existence of a formal asymptotic series solution for large $|\mb x|$ to the problem, we prove its Borel summability (and automatically its asymptoticity) to an actual solution $\mb u$.In special cases motivated by applications we show how the method can be adapted to obtain short-time existence, uniqueness and asymptotic behavior for small $t$,without size restriction on the space variable.
Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…
Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…
The paper shows the summability of formal solutions of some linear q-difference-differential equations by using q-Laplace and q-Borel summation method.
We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $ε$. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so…
In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in \CC^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.
We analyze the detailed time dependence of the wave function $ψ(x,t)$ for one dimensional Hamiltonians $H=-\partial_x^2+V(x)$ where $V$ (for example modeling barriers or wells) and $ψ(x,0)$ are {\em compactly supported}. We show that the dispersive part of $ψ(x,t)$, its asymptotic series in powers of $t^{-1/2}$, is Borel summable. The remainder, the difference between $ψ$ and the Borel sum, is a convergent expansion of the form $\sum_{k=0}^{\infty}g_k Γ_k(x)e^{-γ_k t}$, where $Γ_k$ are the Gamow vectors of $H$, and $γ_k$ are the associated resonances; generically, all $g_k$ are nonzero. For la…
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent Cauchy data. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z). The results are presented in the general framework of moment-PDEs.
We consider the Navier-Stokes initial value problem, $$v_t - \nabla v = -\mathcal{P} [ v \cdot \nabla v \right ] + f, v(x, 0) = v_0 (x), x \in \mathbb{R}^3 $$ where $\mathcal{P}$ is the Hodge-Projection to divergence free vector fields in the assumption that $ | f |_{μ, β} < \infty $ and $| v_0 |_{μ+2, β} < \infty$ for $β\ge 0, μ> 3$, where $$ | {\hat f} (k) | = \sup_{k \in \mathbb{R}^3} e^{β|k|} (1+|k|)^μ| {\hat f} (k) |$$ and ${\hat{f}} (k) = \mathcal{F} [f (\cdot)] (k) $ is the Fourier transform in $x$. By Borel summation methods we show that there exists a classical solution in the form $$…
We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel summability, which allows us to find integral representations of solutions for such PDEs.
We study the Stokes phenomenon via hyperfunctions for the solutions of the 1-dimensional complex heat equation under the condition that the Cauchy data are holomorphic on $\mathbb{C}$ but a finitely many singular or branching points with the appropriate growth condition at the infinity. The main tool are the theory of summability and the theory of hyperfunctions, which allows us to describe jumps across Stokes lines.
We introduce a method of rigorous analysis of the location and type of complex singularities for nonlinear higher order PDEs as a function of the initial data. The method is applied to determine rigorously the asymptotic structure of singularities of the modified Harry-Dym equation $$ H_t + H_y = - {1/2} H^3 + H^3 H_{yyy} : H(y, 0) = y^{-1/2} $$ for small time at the boundaries of the sector of analyticity. Previous work \cite{CPAM}, \cite{invent03} shows existence, uniqueness and Borel summability of solutions of general PDEs. It is shown that the solution to the above initial value problem i…
We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables. The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems.
We study the Stokes phenomenon for the solutions of the 1-dimensional complex heat equation and its generalizations with meromorphic initial data. We use the theory of Borel summability for the description of the Stokes lines, the anti-Stokes lines, jumps across the Stokes lines, and the maximal family of the solutions.
We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is th…
We extend Krylov and Röckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in{\mathbb R}^d,$$ where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\mathcal{C}_q^0((0,T];L^p({\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdo…
This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is then approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the solutions to these SDPs provide bounds on a specified, possibly nonlinea…
