Partial Differential Equations and Applications
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Professor Dajun Guo: a true mathematician and educator
Partial Differential Equations and Applications - - 2020
Higher-order error estimates for physics-informed neural networks approximating the primitive equations
Partial Differential Equations and Applications - - 2023
Large-scale dynamics of the oceans and the atmosphere are governed by primitive
equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of
the PEs is generally challenging. Neural networks have been shown to be a
promising machine learning tool to tackle this challenge. In this work, we
employ physics-informed neural networks (PINNs) to approximate the solutions to
the PEs an...
Blow up of solutions of semilinear wave equations related to nonlinear waves in de Sitter spacetime
Partial Differential Equations and Applications - Tập 3 - Trang 1-10 - 2021
Consider a nonlinear wave equation for a massless scalar field with
self-interaction in the spatially flat de Sitter spacetime. We show that blow-up
in a finite time occurs for the equation with arbitrary power nonlinearity as
well as upper bounds of the lifespan of blow-up solutions. The blow-up condition
is the same as in the accelerated expanding Friedmann–Lemaître–Robertson–Walker
(FLRW) space...
On a hybrid continuum-kinetic model for complex fluids
Partial Differential Equations and Applications - Tập 3 - Trang 1-28 - 2022
In the present work, we first introduce a general framework for modelling
complex multiscale fluids and then focus on the derivation and analysis of a new
hybrid continuum-kinetic model. In particular, we combine conservation of mass
and momentum for an isentropic macroscopic model with a kinetic representation
of the microscopic behavior. After introducing a small scale of interest, we
compute th...
Energy asymptotics for the strongly damped Klein–Gordon equation
Partial Differential Equations and Applications - Tập 3 - Trang 1-12 - 2022
We consider the strongly damped Klein–Gordon equation for defocusing
nonlinearity and we study the asymptotic behaviour of the energy for periodic
solutions. We prove first the exponential decay to zero for zero mean solutions.
Then, we characterize the limit of the energy, when the time tends to infinity,
for solutions with small enough initial data and we finally prove that such
limit is not nec...
Calderón-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient
Partial Differential Equations and Applications - - 2022
Stationary mean-field games with logistic effects
Partial Differential Equations and Applications - Tập 2 - Trang 1-34 - 2021
In its standard form, a mean-field game is a system of a Hamilton-Jacobi
equation coupled with a Fokker-Planck equation. In the context of population
dynamics, it is natural to add to the Fokker-Planck equation features such as
seeding, birth, and non-linear death rates. Here, we consider a logistic model
for the birth and death of the agents. Our model applies to situations in which
crowding incr...
Existence and non-existence of global solutions for a heat equation with degenerate coefficients
Partial Differential Equations and Applications - Tập 3 - Trang 1-16 - 2022
In this paper, the parabolic problem $$u_t - div(\omega (x) \nabla u)= h(t) f(u)
+ l(t) g(u)$$ with non-negative initial conditions pertaining to $$C_b({\mathbb
{R}}^N)$$ , will be studied, where the weight $$\omega $$ is an appropriate
function that belongs to the Muckenhoupt class $$A_{1 + \frac{2}{N}}$$ and the
functions f, g, h and l are non-negative and continuous. The main goal is to
establi...
The Keller–Segel system on bounded convex domains in critical spaces
Partial Differential Equations and Applications - Tập 2 Số 3 - 2021
AbstractConsider the classical Keller–Segel system on a bounded convex domain
$$\varOmega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 . In contrast to previous works it
is not assumed that the boundary of $$\varOmega $$ Ω is smooth. It is shown that
this system admits a local, strong solution for initial data in critical spaces
which extends to a global one provided the data are small enough in this
critica...
Eckhaus instability of stationary patterns in hyperbolic reaction–diffusion models on large finite domains
Partial Differential Equations and Applications - Tập 3 - Trang 1-32 - 2022
We have theoretically investigated the phenomenon of Eckhaus instability of
stationary patterns arising in hyperbolic reaction–diffusion models on large
finite domains, in both supercritical and subcritical regime. Adopting
multiple-scale weakly-nonlinear analysis, we have deduced the cubic and
cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern
amplitude close to critical...
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