Searcharxiv⌕ Search

arXiv subjects

Alexandru F. Radu

Publications and source records attributed to Alexandru F. Radu.

6 recordsLinked to original sources
arXiv2026

Non-unique singular solutions for KP and modified KP equations on $\mathbb T^2$ and $\mathbb R^2$

We construct infinitely many weak singular solutions with zero initial data and compact time support for third- and fifth-order KP-I, KP-II, and their modified counterparts on $\mathbb T^2$ and $\mathbb R^2$. Their nonlinearities are cutoff-independent, absolutely convergent Fourier convolutions. Quadratic solutions belong to $C_tL^p$ for $p<2$ and $C_t(H^{-σ,0}\cap H^{-σ})$ for $σ>0$. Modified solutions belong to $C_tL^p$ for $p<3$. One cubic family lies in $C_tH^α$ for $α<1/3$; another has parabolic Fourier support and lies in $C_tH^{s,0}$ for $s<1/2$ and $C_tH^α$ for $α<1/4$. The exponents $1/3$ and $1/2$ are sharp at the $L^3$ product threshold. For quadratic fifth-order KP on $\mathbb R^2$, the nonuniqueness range is almost sharp. We also construct periodic stationary KP-I and KP-II solutions and prove that $L^2$ is the sharp threshold between singular stationary KP-I solutions and smoothness.

math.AP↗
arXiv2026

Non-unique solutions to the periodic gKdV equation

In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$ \bigcap_{ε> 0} C_t^0 L_x^{k-ε}([0,1] \times \mathbb{T}) $$ and, when $k \ge 3$, it may also be chosen in \[ \bigcap_{ε>0} C_t^0 H_x^{\frac{1}{2} - \frac{1}{k} - ε}([0,1] \times \mathbb{T}) \] attaining identically $0$ initial data. Since our solutions do not lie in $C_t^0 L_x^k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t^0L^1_x$. In the case of KdV this is in fact also sufficient.

math.AP↗
arXiv2026

Uniqueness in Lorentz Spaces of the 2d Navier-Stokes equation

We study uniqueness of mild solutions to the two--dimensional incompressible Navier-Stokes equations on the torus in borderline spatial classes. While Lorentz-space methods yield uniqueness in $C([0,T);L^{2,1}(\mathbb{T}^2))$ via real interpolation and weak $L^2$ control, extending such arguments to larger Lorentz spaces $L^{2,q}$, $1<q<2$, encounters endpoint obstructions. In this paper we prove that uniqueness in $C([0,T);L^{2,q}(\mathbb{T}^2))$ holds provided one assumes a short-time $L^\infty$ smoothing property at every restart time, namely \[ \lim_{δ\downarrow 0}\sup_{t\in(T_0,T_0+δ]}\sqrt{t-T_0}\,\|v(t)\|_{L^\infty(\mathbb{T}^2)}=0, \quad \text{for all } T_0\in[0,T). \] The proof combines the restart mild formulation, the $L^1$ bound for the periodic Oseen kernel of $e^{tΔ}\mathbb{P}\nabla\cdot$, and an explicit Beta-function computation yielding a strict $L^2$ contraction on short intervals. The smoothing assumption is natural in Kato and Koch-Tataru type critical well-posedness frameworks and clarifies how parabolic regularization can replace Lorentz endpoint structure in uniqueness arguments.

math.AP↗
arXiv2025

Intermittent solutions of the stationary 2D surface quasi-geostrophic equation

In this paper we construct non-trivial solutions to the stationary dissipative surface quasi-geostrophic equation on the two dimensional torus which lie strictly below the critical regularity threshold of $\dot{H}^{-1/2}(\mathbb{T}^2)$. Specifically, for any $α< 1/2$ and any dissipation exponent $0 < γ\leq 2$ we construct non-trivial solutions such that $$ u,θ\in \dot{B}^{α-1}_{\infty,\infty}(\mathbb{T}^2) \cap \dot{B}^{α-1}_{2,2}(\mathbb{T}^2). $$ Due to the fact our solutions do not lie in $\dot{H}^{-1/2}(\mathbb{T}^2)$, this requires reinterpreting the notion of a solution. This leads us to formulate the notion of a weak paraproduct solution for the stationary SQG equation. The main new ingredient is the incorporation of intermittency into the construction of the solutions. This allows us to demonstrate non-trivial integrability results for certain fractional derivatives of our solutions. In particular, for highly intermittent solutions, we are able to conclude for every $1 \leq p < 4/3$ we can construct $u$ and $θ$ lying in $L^p(\mathbb{T}^2)$.

math.AP↗
arXiv2022

On the monomial algebra associated to the monomial characters of a finite group

Given a finite group $G$, we study the monomial algebra $R_G$, generated by the monomial characters of $G$. In particular, we note that the integral closure of $R_G$ is contained in the algebra generated by those characters $χ$ for which their associated Artin L-function $L(s,χ)$ is holomorphic at $s_0\in\mathbb C\setminus\{1\}$. Also, we discuss the supercharacter theoretic case.

math.RT↗