Solution of a partial differential equation

A solution of a partial differential equation is a function satisfying the equation in a specified sense and obeying its initial or boundary data.

A classical solution has enough ordinary derivatives for the differential equation and its data to hold pointwise. For the heat equation, $u_t$ and $u_{xx}$ should be continuous in the interior, with the specified initial and boundary traces. A weak solution instead imposes an integrated identity, allowing less regular data; uniqueness must always be stated for the solution class used.

For a semilinear partial differential equation $u_t=\Delta u+F(u)$ with smooth $F$ and smooth fixed boundary data, local boundedness permits positive-time elliptic regularity and parabolic smoothing. If $u_t$ is continuous with values in a Hölder space, elliptic estimates first give continuous $C^{2,\beta }$ spatial regularity for $u$. Time difference quotients then solve a linear parabolic equation with coefficient converging to $F^{\prime }(u)$; smoothing away from the initial time gives $u_t\in C(C^2)$, hence $u\in C^1(C^2)$. It is essential to gain spatial regularity for the time derivative too; $u\in C(C^2)\cap C^1(C^0)$ alone does not imply $C^1(C^2)$.

The notation $C_b^{1,2}([0,\infty )\times \mathbb{R})$ ordinarily denotes functions with continuous time derivative and two continuous space derivatives, with the function and indicated derivatives bounded globally. A different condition, often intended in finite-horizon partial differential equations, is membership in $C_b^{1,2}([0,T]\times \mathbb{R})$ for each finite $T$, allowing bounds to grow with $T$. These conventions must be distinguished in a long-time growth problem: the second permits divergence with time; the first does not. There is no universal notation for the finite-horizon condition, so it should be stated explicitly.

A weak solution satisfies an integrated identity in which derivatives are transferred to test functions. It can therefore exist with fewer classical derivatives.

For a partial differential equation $F(x,u,Du,D^{2}u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\varphi (x_0),D^2\varphi (x_0))\le 0$ whenever a smooth test function $\varphi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\mathrm{tr}{D}^{2}u$, these conditions are respectively $\Delta \varphi \ge 0$ and $\Delta \varphi \le 0$. They make sense even when $u$ has no classical second derivatives.

Let a continuous function satisfy both viscosity inequalities for the Laplace operator. On any relatively compact ball, take its harmonic replacement $h$ with matching boundary values. If $u-h$ has a positive maximum, the function $h+\epsilon (R^2-|x-a{|}^2)$ still lies below $u$ somewhere for sufficiently small positive $\epsilon$, while matching it on the boundary. At an interior maximum of their difference it is a valid upper test, but its Laplacian is $-2n\epsilon <0$, a contradiction. The opposite perturbation excludes a negative minimum. Thus $u=h$ throughout the ball and is a smooth harmonic function. A locally defined test is extended near its touching point using a smooth cutoff function when global tests are required.

A weak-strong uniqueness principle says that a weak solution with the same data as a sufficiently regular classical solution agrees with that solution throughout its interval of regular existence, within the specified weak class. The conclusion is class-dependent: all bounded weak solutions of a prescribed $C^1$ linear transport field satisfy it, while nonlinear conservation laws require an admissibility condition such as an entropy solution condition.

For $u_{tt}+L(t)u=f$ with homogeneous Dirichlet boundary conditions, smooth uniformly elliptic principal coefficients, and initial data $({\psi }_0,{\psi }_1)\in H_0^1(U)\times L^2(U)$, a space-time $H^1$ weak solution has zero lateral trace, trace $u(0)={\psi }_0$, and satisfiesfor every space-time $H^1$ test function with zero lateral trace and $v(T)=0$. The velocity initial condition is encoded by the last term; an arbitrary space-time $H^1$ function need not have an $L^2$ trace of its time derivative. The equation itself gives $u_{tt}\in L^2(0,T;H^{-1}(U))$, so $u_t$ has a continuous $H^{-1}$ representative.

A weak formulation multiplies a differential equation by a test function, integrates, and uses integration by parts to move derivatives away from the unknown function. Natural boundary conditions appear in the resulting boundary terms.

The Galerkin method seeks an approximate solution in a finite-dimensional subspace and requires the equation's residual to be orthogonal to that subspace. Uniform energy estimates and compactness can then produce a weak solution as the subspaces become dense.

The Petrov-Galerkin method chooses an approximate solution $u_h$ in a finite-dimensional trial vector space $V_h$ and tests a weak formulation against a possibly different test space $W_h$. For a bilinear form $a$ and a linear functional $\ell$, it imposes $a(u_h,v_h)=\ell (v_h)$ for every $v_h\in W_h$. Taking $W_h=V_h$ recovers the usual Galerkin method. Different test functions can alter stability and accuracy; the Galerkin orthogonality is then against the test space and need not be an orthogonal projection in an energy norm.

A Fourier truncation retains a finite set of boundary-compatible Fourier modes and projects the PDE onto their span by the Galerkin method. Nonlinear products can generate unretained harmonics, so their coefficients are discarded by projection rather than declared zero pointwise. Agreement with a perturbation expansion through cubic order requires retention of every quadratically forced mode that feeds back on the critical mode at cubic order.

// Destination: dynamical-systems.bigb

If $X_0$ embeds compactly into $X$ and $X$ embeds continuously into $X_1$, then boundedness in $L^p(0,T;X_0)$ together with a suitable time-derivative bound in $L^q(0,T;X_1)$ makes a family relatively compact in $L^p(0,T;X)$. A standard case is

If $u\in L^{\infty }(0,T;H)$ and $u_t\in L^q(0,T;V^{\prime })$ for a dense continuous embedding $V\hookrightarrow H$ and some $q>1$, then $u$ has a representative in $C([0,T];H_{\mathrm{weak}})$. Indeed, its pairing with each element of a dense subset of $V$ is absolutely continuous, and the uniform $H$ bound extends continuity to every element of $H$.

If $u:[0,T]\to H$ is weakly continuous and an energy equality makes $t\mapsto \Vert u(t){\Vert }_H$ continuous, then $u$ is strongly continuous. This follows from the Radon-Riesz theorem applied whenever $t_n\to t$.