**3**

votes

**0**answers

40 views

### Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...

**2**

votes

**0**answers

39 views

### Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation
$$
\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\
u(0)=u_{0}.
$$
with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...

**0**

votes

**2**answers

90 views

### Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ ...

**-2**

votes

**0**answers

35 views

### Numerical method of lines for solving PDEs [closed]

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...

**0**

votes

**0**answers

46 views

### Solution of parabolic PDE system [closed]

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$.
\begin{equation}
\begin{cases}
\frac{\partial}{\partial ...

**1**

vote

**0**answers

41 views

### Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...

**1**

vote

**0**answers

43 views

### References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case:
\begin{equation}
Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + ...

**0**

votes

**1**answer

51 views

### Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following:
Suppose we have a uniformly parabolic equation with holder ...

**5**

votes

**0**answers

89 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**0**

votes

**1**answer

88 views

### Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials.
If $\Gamma(t,x)$ is the fundamental ...

**2**

votes

**0**answers

77 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...

**2**

votes

**0**answers

47 views

### Solve a PDE related to free boundary problem

I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...

**3**

votes

**1**answer

139 views

### Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...

**-2**

votes

**2**answers

108 views

### Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?

**9**

votes

**1**answer

196 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

**1**

vote

**1**answer

105 views

### Decay of Solutions to the Heat equation

Consider the heat equation
$$ (\partial_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.
Consider the semigroup ...

**0**

votes

**1**answer

158 views

### When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...

**0**

votes

**0**answers

30 views

### Reference for a regularity result for linear parabolic equations under mixed Neumann-Dirichlent boundary condition

Could someone please help me to find a precise reference for $L^{p}%
-$regularity results for linear parabolic equations under mixed
Neumann-Dirichlet conditions? More precisely, I would like to have ...

**1**

vote

**1**answer

204 views

### Quadratic PDE Systems

(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)
I have a problem that leads me to the following quadratic system of PDEs:-
$
c_1 ...

**1**

vote

**0**answers

46 views

### Well-posedness of a certain linear Cauchy-problem

I am interested in solutions to the linear Cauchy problem
$$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, ...

**4**

votes

**0**answers

80 views

### Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...

**1**

vote

**0**answers

59 views

### What's Known About the Green's Function to the 1D Diffusion Equation with Position-dependent Diffusion Coefficient?

Consider a one-dimensional diffusion equation
$$
C(x) \partial_t \Phi(t,x) = \partial_x^2 \Phi(t,x),
$$
on the interval $[0,1]$. The function $C(x)$ has a pole of order 1 at $x=0$ and a pole of finite ...

**2**

votes

**1**answer

100 views

### Heat transfer: boundary conditions with fluid velocity

The following equation is considered:
$$
\frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f.
$$
I have difficulties in formulating boundary conditions for this equation.
If ...

**1**

vote

**1**answer

108 views

### Interpolation and embeddings for parabolic function spaces

I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...

**1**

vote

**1**answer

99 views

### How to model a time-discrete heat equation on a graph?

I would like to know how one can set up a time-discrete model for the heat equation on a graph?
Is this problem using the heat kernel equation on a graph?

**0**

votes

**0**answers

72 views

### Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...

**0**

votes

**0**answers

55 views

### Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges?
Here is an example: How to solve the equation ...

**3**

votes

**1**answer

490 views

### Questions on the proof of the Serrin condition for the regularity of Navier-Stokes equations and related issues for the incompressible Euler equation

Edit: The question has been substantially modified from the original one. The original question (see below) concerned with rigorously justifying the proof of the Serrin condition. These questions have ...

**1**

vote

**1**answer

104 views

### Existence of the solution of a linear parabolic pde

Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...

**7**

votes

**1**answer

322 views

### About the convergence rate for an approximation to the heat kernel

Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} ...

**2**

votes

**1**answer

226 views

### A comparison principle for parabolic equation

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...

**3**

votes

**0**answers

69 views

### Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...

**0**

votes

**0**answers

78 views

### Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...

**0**

votes

**0**answers

62 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**2**

votes

**0**answers

78 views

### Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...

**0**

votes

**0**answers

54 views

### Interpolation with time continuity

If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in ...

**2**

votes

**0**answers

62 views

### Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...

**3**

votes

**1**answer

112 views

### Checking initial data in parabolic PDE with no control on time derivative

It is possible to define a weak solution of a parabolic PDE
$$u_t - Au = f$$
$$u(0) = u_0$$
as $u \in L^2(0,T;H^1)$ such that
$$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega ...

**3**

votes

**0**answers

147 views

### Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < ...

**1**

vote

**0**answers

113 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**0**

votes

**0**answers

53 views

### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...

**3**

votes

**1**answer

163 views

### parabolic PDE with almost-monotone elliptic operator, existence results?

Are there any existence results for parabolic PDE of the type $$u_t - Au = f$$ in some Gelfand triple setting ($V \subset H \subset V^*$) with $A$ an operator that it is not quite monotone but close: ...

**1**

vote

**0**answers

102 views

### Weak periodic solution of parabolic PDE

Take
$$
u_t(t) + A(t)u(t) = f(t),
$$
$$
u(0) = u(T),
$$
where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...

**1**

vote

**0**answers

95 views

### Proving solution to linear parabolic PDE is convex with negative third derivative

I have a PDE in $g(y,t)$ of the form
\begin{equation}
a\frac{\partial^2g}{\partial y^2}y^2-b\frac{\partial g}{\partial y}y -rg + \frac{\partial g}{\partial t} - c = 0
\end{equation}
in which $a$, $b$, ...

**1**

vote

**1**answer

112 views

### What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''?

What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to
$Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1.
This ...

**1**

vote

**0**answers

138 views

### Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
...

**1**

vote

**0**answers

74 views

### monotone parabolic systems, convex variational structure and Legendre transform

The context:
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & ...

**5**

votes

**2**answers

309 views

### If a PDE have a unique classical solution, must it have a unique viscosity solution?

If a PDE have a unique classical solution, must it have a unique viscosity solution?
The particular problem I am interested in is parabolic, but I would be interested in the general case.
A short ...

**1**

vote

**1**answer

113 views

### Differences between parabolic operators of second order and higher order

Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly ...

**1**

vote

**1**answer

80 views

### Solution of a partial differential equation containing a Fourier series

Given the following PDE:
$$\partial_t\Psi(x,t)=\partial_{xx}\Psi(x,t)+k\partial_x\Psi(x,t)+g(x,t)-\beta\Psi(x,t)=0$$
where:
...