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We study the Cauchy problem for the chemotaxis Navier-Stokes equations and the Keller-Segel-Navier-Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity…

Analysis of PDEs · Mathematics 2023-01-04 Gael Yomgne Diebou

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of…

Probability · Mathematics 2018-03-15 Franco Flandoli , Dejun Luo

The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain…

Mathematical Physics · Physics 2015-05-20 Marco Bertola , Thomas Bothner

The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…

Classical Analysis and ODEs · Mathematics 2016-08-03 Myong-Ha Kim , Guk-Chol Ri , Gum-Song Choe , Hyong-Chol O

The existence theory for solutions of the linearized field equations for causal variational principles is developed. We begin by studying the Cauchy problem locally in lens-shaped regions, defined as subsets of space-time which admit…

Mathematical Physics · Physics 2021-01-25 Claudio Dappiaggi , Felix Finster

A kinetic model with flexible velocities is presented for solving the multi-component Euler equations. The model employs a two-velocity formulation in 1D and a three-velocity formulation in 2D. In 2D, the velocities are aligned with the…

Fluid Dynamics · Physics 2026-02-17 Shashi Shekhar Roy , S. V. Raghurama Rao

In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is…

Analysis of PDEs · Mathematics 2018-05-25 Misha Vishik

We consider some parabolic equations which are model problems for a variety of nonlinear generalizations to the Black-Scholes equation of mathematical finance. In particular, we prove local well-posedness for the Cauchy problem with initial…

Analysis of PDEs · Mathematics 2018-12-17 Daniel Oliveira da Silva , Kamilla Igibayeva , Adelina Khoroshevskaya , Zhanna Sakayeva

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of…

Analysis of PDEs · Mathematics 2014-10-24 Zhitao Zhang , Yimin Sun

In this paper, we focus on the existence of strong solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation. We construct a new combination of Bourgain space and Lebesgue space where linear and…

Analysis of PDEs · Mathematics 2023-06-06 Chenlu Zhang , Huaqiao Wang

This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally…

Analysis of PDEs · Mathematics 2025-04-02 Thomas Alazard , Chengyang Shao , Haocheng Yang

This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key…

Analysis of PDEs · Mathematics 2026-02-02 Bing Yuan , Rong Zhang , Peng Zhou

In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any…

Classical Analysis and ODEs · Mathematics 2024-11-12 Luca Tanganelli Castrillón

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…

Analysis of PDEs · Mathematics 2007-05-23 Francois James , Simona Mancini , Francois Bouchut

This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this…

Analysis of PDEs · Mathematics 2007-05-23 Veronica Felli , Matthias Schneider

We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…

Analysis of PDEs · Mathematics 2020-10-30 Olga Rozanova

We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong…

Probability · Mathematics 2020-09-23 Martina Hofmanová , Rongchan Zhu , Xiangchan Zhu

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

Analysis of PDEs · Mathematics 2011-04-07 Clemens Hanel

In this paper, we present some results for existence of global solutions and attractivity for mulidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and Banach fixed…

Classical Analysis and ODEs · Mathematics 2017-09-08 H. T. Tuan , Adam Czornik , J. Nieto , M. Niezabitowski

We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization…

Differential Geometry · Mathematics 2015-07-21 Hong Huang
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